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Envision Math Accelerated Grade 7 Volume 1 Student Edition Chapter 6 Solving Problems Using Equations And Inequalities

Question. Find out the relationship between the shapes.

Given: Three clues are given.

We have to find out the relationship between the shapes.

First, we will look for that shape which is common on both the left side and right side and then we will eliminate it and find out the relationship between the remaining shapes.

Let, C represents a circle, T represent a triangle and S represents a square.

Therefore, we get the following clues as

​4C  + 1T = 2S + 1T

3T + 2C = 2C + 1S

1S + 2C = 6T

​From the first clue we have

4C + 1T = 2S + 1T

4C + T−T = 2S + T − T (Subtracting T from both sides)

4C = 2S

\(\frac{4}{2}\)C=\(\frac{2}{2}\)S   (Dividing 2 from both sides)

S = 2C ———-(1)

From the second clue we have

3T+2C=2C+1S

3T + 2C − 2C = 2C + S − 2C     (Subtracting 2C from both sides)

S = 3T ———–(2)

From equation (1) and (2)

We have, S = 2C  and  S =   3T

Therefore, 2C = 3T———(3)

From equations (1), (2), and (3)

We get the relationship between the shapes

S=2C

S=3T

2C=3T

The relationship between the shapes are as follows: S = 2C, S = 3T, 2C = 3T

Envision Math Accelerated Grade 7 Chapter 6 Exercise 6.2 Answer Key

Question. Use properties of equality to solve the given equations

4C + 1T = 2S + 1T

3T + 2C = 2C + 1S

1S + 2C = 6T

We have to use properties of equality to reason about the equations.

Let, C represents a circle, T represent a triangle and S represents a square.

Therefore, we get the following clues as

​4C + 1T = 2S + 1T

3T + 2C = 2C + 1S

1S + 2C = 6T

​From the first clue we have

4C + 1T = 2S + 1T

4C + T − T = 2S + T−T             (Subtracting T from both sides)

4C = 2S

\(\frac{4}{2}\)C =  \(\frac{2}{2}\) S                      (Dividing 2 from both sides)

S = 2C ———-(1)

From the second clue we have

3T + 2C = 2C + 1S

3T + 2C − 2C = 2C + S − 2C     (Subtracting 2C from both sides)

S = 3T ———–(2)

From equation (1) and (2)

We have, S = 2C and S = 3T

Therefore, 2C = 3T——–(3)

From equations (1), (2), and (3)

We get the relationship between the shapes:

S = 2C

S = 3T

2C = 3T

We used the subtraction and division properties of equality to solve the given equations.

Envision Math Grade 7 Chapter 6 Solving Equations And Inequalities Exercise 6.2 Solutions

Question. A shape equation C represents a circle, T represent a triangle and S represents a square.

Given: A shape equation is given.

Relationship between shapes:

S = 2C

S = 3T

2C = 3T

We have to complete the equation only using triangles.

Let, C represents a circle, T represent a triangle and S represents a square.

Therefore, we get

4C + S = ?

From solving and discussing exercise 1 we have the following relationship between the shapes.

S = 2C

S = 3T

2C = 3T

Therefore, ​ 4C = 6T

S = 3T

​Hence, we get

4C + S = 6T + 3T = 9T

Therefore, the complete equation is 4C + S = 9T

The complete equation is 4C + S = 9T

Question. Write a two-step equation is similar to solving a one-step equation.

We have to explain how solving a two-step equation is similar to solving a one-step equation.

Solving a two-step equation is similar to solving a one-step equation because they both use properties of equality.

The properties of equality can be applied the same way when solving two-step equations as when solving one-step equations.

Both methods use properties of equality to determine the value of the variable or to make the equation balance.

Solving a two-step equation is similar to solving a one-step equation because they both use properties of equality.

Question. Andrew rents bowling shoes for $4 and he bowls 2 games. Find out the cost of each game.

Given:

Andrew rents bowling shoes for $4 and he bowls 2 games.

The total money he spent is $22

Given bar diagrams are:

We have to complete the bar diagrams and find out the cost of each game.

The total money Andrew spent is $22

He rents the bowling shoes for $4 and played 2 games.

Therefore, we get the following bar diagrams:

The total cost is $22

Therefore, 4 + 2b = 22

4 + 2b − 4 = 22− 4     (Subtracting 4 from both sides)

2b = 18

\(\frac{2}{2}\) = \(\frac{18}{2}\)   (Dividing 2 from both sides)

b = 9

Therefore, the cost of each game is $9.

The complete bar diagrams are:

The cost of each game is $9.

Given:

Kristy ran 24 laps.

The total distance traveled by Kristy is 29.6 kilometers

She walked 0.2 km to the presentation table.

We have to determine the distance of each lap.

First, we will determine the total distance she traveled in 24 laps and then we will divide it by 24 to get the distance of each lap.

The total distance traveled by Kristy is 29.6 kilometers

She walked 0.2 km to the presentation table.

Therefore, total distance she covered in  24 laps = 29.6 − 0.2 = 29.4 km     (Subtraction)

Distance covered in 1 lap \(=\frac{29.4}{24}\)        (Division)

=1.225

Hence, the distance of each lap is 1.225 km

The two steps we used for solving the situation are subtraction and division.

The distance of each lap is 1.225 km. The two steps we used for solving the situation are subtraction and division.

Solving Problems Using Equations And Inequalities Grade 7 Exercise 6.2 Envision Math

Question. Solve the equation by using subtraction and division.

We have to explain what were the two steps we used to solve the equation.

The total cost is $22

Therefore, 4 + 2b = 22

4 + 2b−4 = 22 − 4       (Subtracting 4 from both sides)

2b = 18

\(\frac{2}{2}b\)=\(\frac{18}{2}\)              (Dividing 2 from both sides)

b = 9

Therefore, the cost of each game is $9.

We can see that the two steps we used to solve the equation are subtraction and division.

The two steps we used to solve the equation are subtraction and division.

Question. Write a two-step equation is similar to solving a one-step equation.

We have to explain how solving a two-step equation is similar to solving a one-step equation.

Solving a two-step equation is similar to solving a one-step equation because they both uses properties of equality.

The properties of equality can be applied the same way when solving two-step equations as when solving one-step equations.

Both methods use properties of equality to determine the value of the variable or to make the equation balance.

Two-step equations can be solved in two steps by using two different properties of equality.

Solving a two-step equation is similar to solving a one-step equation because they both use properties of equality.

Question. Use bar diagram for solve 4x – 3 = 13.

Given:

Bar diagram for 4x − 3 = 13 is given

we have to solve the bar diagram for x

4x−  3 = 13

4x − 3 + 3 = 13 + 3    (Adding 3 to both sides)

4x = 16

\(\frac{4x}{4}\)=\(\frac{16}{4}\) (Dividing 4 from both sides))

x = 4

The value of x is 4.

We used addition and division to determine the value of x from the bar diagram.

The value of x is 4.

Envision Math Grade 7 Exercise 6.2 Solution Guide

Question. Solve the equation 6p – 12 = 72 and said p = 14.

Given:

Clara solved a problem 6p − 12 = 72 and said p = 14.

We have to check whether Clara is correct or not.

First, we will substitute the value of p and calculate the left-hand side of the equation.

Given equation: 6p − 12 = 72 and p = 14

The left-hand side of the equation is 6p − 12

Substituting the value of p = 14 in 6p − 12 we get

6 (14) − 12      (Multiply)

=  84 − 12        (Subtract)

=  72

=   RHS

Since the left-hand side of the equation is equal to the right-hand side of the equation.

Therefore, the value of p is correct.

And hence, Clara is correct.

As the left-hand side of the equation is equal to the right-hand side of the equation. therefore, the value of p is correct. And hence, Clara is correct.

Question. Clyde is baking, and the recipe requires 1\(\frac{1}{3}\) cups of flour. Clyde has 2 cups of flour, but he is doubling the recipe to make twice as much. Find out how much more flour Clyde needs.

Given:  Clyde is baking, and the recipe requires  1\(\frac{1}{3}\)  cups of flour. Clyde has 2 cups of flour, but he is doubling the recipe to make twice as much.

We need to find out how much more flour Clyde needs.

Also, we need to write an equation to represent the problem.

Let c represent the amount of flour Clyde needs.

The number of cups required for the recipe is 1\(\frac{1}{3}\)

Clyde has two cups of flour.

He is doubling the recipe to make twice as much.

Here, c represents the amount of flour Clyde needs.

The equation that represents the problem will be

2 + c = 2 × 1\(\frac{1}{3}\)

The equation that represents the problem will be 2 + c = 2 × 1\(\frac{1}{3}\)

Given: Clyde is baking, and the recipe requires 1\(\frac{1}{3}\)cups of flour.

Clyde has 2 cups of flour, but he is doubling the recipe to make twice as much.

We need to find how much more flour does Clyde need. We need to solve the equation formed.

The equation formed will be 2 + c = 2 × 1\(\frac{1}{3}\)

Solving the given equation, we get

2 + c = \(\frac{8}{3}\)

c = \(\frac{8}{3}-2\)

c = \(\frac{8-6}{3}\)

c = \(\frac{2}{3}\)

\(\frac{2}{3}\) cups of flour Clyde needs to complete the recipe.

How To Solve Exercise 6.2 In Envision Math Grade 7

Question. Four times a number, n, added to 3 is 47. Write an equation that you can use to find the number.

Given: Four times a number, n, added to 3 is 47.

We need to write an equation that you can use to find the number.

Let the unknown number be n

The four times of the unknown number is added to three.

The result obtained is 47

The equation will be

4n + 3 = 47

The equation that we can use to find the number will be, 4n + 3 = 47

Given: Four times a number, n, added to 3 is 47.

We need to find the number represented by n

The equation formed from the given data is

4n + 3 = 47

Solving the equation, we get

​4n + 3 = 47

4n = 47 − 3

4n = 44

n  =  \(\frac{44}{4}\)

n = 11

The number represented by n is 11.

Envision Math 7th Grade Exercise 6.2 Step-By-Step Solutions

Question. Solve the equation 4x – 12 = 16.

We need to use the bar diagram to help you solve the equation  4x − 12 = 16.

The given equation is 4x − 12 = 16.

Solving the equation, we get

​4x − 12 = 16

4x = 16 + 12

4x = 28

x =  \(\frac{28}{4}\)

x = 7
​
Solving the given using the bar diagram, we get

​16 + 12 = 4x

28 = 4x

x = \(\frac{28}{4}\)

x = 7
​
The value of  x = 7

Question. Solve the given equation.

We need to complete the steps to solve the given equation.

We need to place the missing numbers in the given equation to solve it.

Thus, we get

\(\frac{1}{5}\)t+2−2 = 17−2

\(\frac{1}{5}\)t  = 15

5×\(\frac{1}{5}\)t = 5 × 15

t = 75

The value of  t = 75

Envision Math Accelerated Chapter 6 Exercise 6.2 Solving Equations And Inequalities

Question. Use the bar diagram to write an equation. Then solve for x.

We need to use the bar diagram to write an equation. Then solve for x.

The bar diagram given is

The equation formed from the bar diagram is

x + x + x = 7 + 5

Solving the given equation, we get

​x + x + x = 7 + 5

3x = 12

x = \(\frac{12}{3}\)

x = 4

The value of x = 4

Question. While shopping for clothes, Tracy spent $38 less than 3 times what Daniel spent. Find how much Daniel spent.

Given that, While shopping for clothes, Tracy spent $38 less than 3 times what Daniel spent.

We need to write and solve an equation to find how much Daniel spent.

Let x represent how much Daniel spent. Also, the amount Tracy spent is $10.

The equation represented by the given situation will be

3x − 38 = 10

Solving the given equation, we get

​3x − 38 = 10

3x = 10 + 38

3x = 48

x = \(\frac{48}{3}\)

x = 16

Daniel spent $16

Envision Math Accelerated Grade 7 Chapter 6 Exercise 6.2 Answers

Question. Solve the equation 0.5p – 3.45 = -1.2.

We need to solve the equation 0.5p − 3.45 = −1.2.

The given equation is 0.5p−3.45 = −1.2.

Solving the given we get

0.5p – 3.45 = -1.2.

0.5p = -1.2 + 3.45

0.5p = 2,25

p = \(\frac{2.25}{0.5}\)

p = \(\frac{22.5}{5}\)

p = 4.5

The value of p = 4.5

Question. Solve the equation \(\frac{n}{10}\) + 7 = 10

We need to solve the equation \(\frac{n}{10}\)+7 = 10

The given equation is n \(\frac{n}{10}\)+7 = 10

Solving the given we get

\(\frac{n}{10}\) + 7 = 10

\(\frac{n}{10}\) = 10− 7

\(\frac{n}{10}\) = 3

n = 3 × 10

n = 30

The value of n = 30

Question. A group of 4 friends went to the movies. In addition to their tickets, they bought a large bag of popcorn to share for $6.25.

Given:

A group of 4 friends went to the movies.

In addition to their tickets, they bought a large bag of popcorn to share for $6.25.

The total was $44.25

To find/solve

Write and solve an equation to find the cost of one movie ticket, m.

Use an equation to determine the cost of one ticket.

4m + 6.25 = 44.25

4m = 38

m = 9.50

The cost of one ticket is $9.50.

The equation is 4m + 6.25 = 44.25, The cost of one ticket is $9.50.

Given:

A group of 4 friends went to the movies.

In addition to their tickets, they bought a large bag of popcorn to share for $6.25.

The total was $44.25.

To find/solve

Draw a model to represent the equation.

4m + 6.25 = 44.25

4m = 38

m = 9.50

The bar diagram represents the equation

The bar diagram represents the equation

Given:

Oliver incorrectly solved the equation  2x + 4 = 10.

He says the solution is x = 7.

To find/solve

What is the correct solution?

The correct solution is

2x + 4 = 10

2x = 6

x = 3

The correct solution is x = 3.

Given:

Oliver incorrectly solved the equation 2x+4=10. He says the solution is x=7.

To find/solve

 What mistake might Oliver have made?

An expression is a finite combination of symbols that is well-formed according to rules that depend on the context.

Oliver did not use the inverse relationship of the equation.

He did not apply the properties of equality in order to balance the equation.

He added 4 on the right side of the equation instead of subtracting.

Oliver did not use the inverse relationship of the equation. He did not apply the properties of equality in order to balance the equation. He added 4 on the right side of the equation instead of subtracting.

Envision Math Grade 7 Chapter 6 Practice Problems Exercise 6.2

Question. What two properties of equality do you need to use to solve the equation?

An expression is a finite combination of symbols that is well-formed according to rules that depend on the context.

The two properties of equality that is needed to solve for the equation are the addition property of equality and the division property of equality.

Question. Find the solution of the equation 4.9x – 1.9 = 27.5

Given:

Use the equation 4.9x − 1.9 = 27.5

To find/solve

The solution is x =?

Determine the solution:
​
​4.9x −1.9 = 27.5

4.9x = 29.4

x = 29.4/4.9

x = 6

The solution is x = 6.

Question. At a party, the number of people who ate meatballs was 11 fewer than of the total number of people. Five people ate meatballs.

Given:

At a party, the number of people who ate meatballs was 11 fewer than of the total number of people.

Five people ate meatballs.

We form the equation as:

5=\(\frac{1}{3}\)x−11

Now we write a one-step equation that has the same solution:

\(\frac{1}{3}\)x = 16

Therefore, this is the required one-step equation that has the same solution.

Therefore, the required one-step equation with the same solution is \(\frac{1}{3}\)x=16

Envision Math Accelerated 7th Grade Exercise 6.2 Key

Question. In a week, Tracy earns $12.45 less than twice the amount Kalya earns. Tracy earns $102.45.

Given:

In a week, Tracy earns $12.45 less than twice the amount Kayla earns. Tracy earns $102.45.

We form the equation as:

102.45 = 2x − 12.45

Now we solve the equation

​2x = 102.45 + 12.45

2x = 114.9

x = \(\frac{114.9}{2}\)

x = 57.45

Therefore, Kayla earns $57.45.

Therefore, Kayla earns $57.45

Question. Solve the equation 2x + 4\(\frac{1}{5}\) = 9

Given: 2x+4\(\frac{1}{5}\) = 9

We solve the equation:

2x + 4\(\frac{1}{5}\) = 9

Now we use inverse operations and the subtraction property of equality to isolate the term with the variable.

​2x+ \(4 \frac{1}{5}-4\frac{1}{5}\)

= 9− 4\(\frac{1}{5}\)

2x = 4\(\frac{1}{5}\)

2x = \(\frac{41}{5}\)

Now we use inverse operations and the division property of equality to isolate the variable.

\(\frac{2x}{2}\) = \(\frac{41}{5}\).\(\frac{1}{2}\)

x = 4.1

Therefore, the required solution is x = 4.1

Envision Math Accelerated Grade 7 Volume 1 Chapter 6 Solving Problems Using Equations And Inequalities

Question. Six friends go white-water rafting. The total cost for the adventure is $683.88, including a $12 fee per person to rent flotation vests. Marcella says they can use the equation 6r + 12 = 683.88 to find the rafting cost, r, per person, Julia says they need to use the equation 6(r + 12) = 683.88.

Given:

Six friends go white-water rafting.

The total cost for the adventure is $683.88, including a $12 fee per person to rent flotation vests.

Marcella says they can use the equation 6r  + 12 = 683.88 to find the rafting cost, r, per person, Julia says they need to use the equation 6 (r + 12)  =  683.88

To find/solve

Question. Whose equation accurately represents the situation? Construct an argument to support your response.

The distributive property tells us how to solve expressions in the form of a (b  +  c).

The distributive property is sometimes called the distributive law of multiplication and division.

The equation given by Julia shows the correct representation of the situation.

In her equation she showed that the $12 fee for the flotation vests rental must be adequately distributed to the number of persons who will be using it.

Julia has an accurate representation of the equation.

Given:

Six friends go white-water rafting.

The total cost for the adventure is $683.88, including a $12 fee per person to rent flotation vests.

Marcella says they can use the equation 6r +12=683.88 to find the rafting cost, r, per person, Julia says they need to use the equation 6(r+12) = 683.88

To find/solve

Question. What error in thinking might explain the inaccurate equation?

In the first equation, it shows that the flotation vests rental is added to the rafting cost of all six friends.

Marcella may have thought that the $12 flotation vests fee is for all and not per person.

Marcella might be thinking that the flotation vest rental fee covers the whole group as one and not individually.

Question. How does the Distributive Property help you solve equations?

Solving equations using distributive property involves the quantities and variables found in the given situation.

It will enable to make difficult problems easier by breaking down the factor as a difference of two numbers which indicates the quantities stated in the problem.

Distributive property helps solve equations by multiplying the two terms in the parenthesis by the term outside the parenthesis.

Envision Math Accelerated Grade 7 Chapter 6 Exercise 6.3 Answer Key

Question. Can the equation 32x + 2.32 = 114.24 be used to find the original cost of each figurine in the problem above.

Given:

Can the equation 32x  +  2.32  =  114.24 be used to find the original cost of each figurine in the problem above.

To find/solve

Explain.

− \(\frac{1}{2}b\) +\( \frac{6}{2}\) = 5

− \(\frac{1}{2}b\) + 3 = 5

− \(\frac{1}{2}b\) − 3 + 3 = 5−3

− \(\frac{1}{2}b\) =2

− 2 (\(\frac{-1}{2}b\)) = (2) − 2

b = −4

0.4x − 0.18 = 9.2

0.4x = 9.2 + 0.18

x = 23.45

−4p = 848 = 44

-4p = 44−848

−4p = −804

P = 201

Solution

1)  B  =  −4

2)  X  =  23.45

3)  P  =  201

Question. How does the Distributive Property help you solve equations?

Given:

To find/solve

How does the Distributive Property help you solve equations?

Solving equations using distributive property involves the quantities and variables found in the given situation.

It will enable to make difficult problems easier by breaking down the factor as a difference of two numbers which indicates the quantities stated in the problem.

Distributive property helps solve equations by multiplying the two terms in the parenthesis by the term outside the parenthesis.

Question. How are the terms in parentheses affected when multiplied by a negative coefficient when the Distributive Property is applied?

Given

Statement

To find/solve

How are the terms in parentheses affected when multiplied by a negative coefficient when the Distributive Property is applied?

The terms inside the parenthesis are affected if a negative coefficient is multiplied using the distributive property.

If the terms inside the parenthesis are negative then the product will be positive and if the terms inside the parenthesis are positive the product will be negative.

The signs will be changed if the number outside has a negative coefficient.

Signs inside the parenthesis will be changed if the term outside has a negative coefficient.

Question. How can an area model help you set up an equation for a problem situation?

Given

Statement

To find/solve

How can an area model help you set up an equation for a problem situation?

Using an area model can easily visualize how to represent an equation for a given problem.

5 (x + 2) = 35

It can easily visualize the quantities involved in the problem.

Envision Math Grade 7 Chapter 6 Solving Equations And Inequalities Exercise 6.3 Solutions

Question. A family of 7 bought tickets to the circus. Each family member also bought a souvenir that cost $6. Total amount they spent was $147. How much did one ticket cost?

Given:

A family of 7 bought tickets to the circus.

Each family member also bought a souvenir that cost $6.

The total amount they spent was $147

To find/solve

How much did one ticket cost?

Determine the amount of one ticket

​7(x + 6) = 147

7x + 42 = 147

7x + 42 − 42 = 147 − 42

7x = 105

x = 15
​
One ticket costs $15.

Question. David says that the original price of the shorts was $41. Does his answer seem reasonable? Defend your answer by writing and solving an equation that represents the situation.

Given:

David says that the original price of the shorts was $41.

To find/solve

Does his answer seem reasonable? Defend your answer by writing and solving an equation that represents the situation.

David did not get the correct original price of the shorts and the price is not reasonable.

The correct equation that represents the situation is

​1/4 (x + 18) = 10.25

1/4x + 18/4 = 10.25

1/4x + 4.5 = 10.25

1/4x = 5.75

x = 23
​
The original price of the shorts is $23.

Solving Problems Using Equations And Inequalities Grade 7 Exercise 6.3 Envision Math

Question. Which of the following shows the correct use of the Distributive Property when solving \(\frac{1}{3}\)(33 – x) = 135.2 the equation.

We need to find which of the following shows the correct use of the Distributive Property when solving \(\frac{1}{3}\)(33-x) = 135.2

The given expression is \(\frac{1}{3}\)(33 − x)  =  135.2

Using the distributive property, the expression becomes

\(\frac{1}{3}\)(33 − x) = 135.2

\(\frac{1}{3}\)×33 − \(\frac{1}{3}\)x = 135.2

\(\frac{1}{3}\).33 −  \(\frac{1}{3}\).x = 135.2

The correct use of the given expression using the distributive property will be (D) 1

\(\frac{1}{3}\).33-\(\frac{1}{3}\).x = 135.2

Question. Use the Distributive Property to solve \(\frac{1}{8}\)(p + 24) = 9 equation.

We need to use the Distributive Property to solve the given equation.

The given equation is \(\frac{1}{8}\)(p + 24) = 9

The given expression is \(\frac{1}{8}\)(p+24) = 9

Using the distributive property, we get

\(\frac{1}{8}\)(p + 24) = 9

\(\frac{p}{8}\) + \(\frac{24}{8}\)= 9

\(\frac{p}{8}\)+3 = 9

\(\frac{p}{8}\) = 9−3

\(\frac{p}{8}\) = 6

p = 6 × 8

p = 48

The value of p=48

Question. Use the Distributive Property to solve the \(\frac{2}{3}(6a+9)\) = 20.4 equation.

We need to use the Distributive Property to solve the given equation.

The given equation is \(\frac{2}{3}(6a+9)\) = 20.4

The given expression is \(\frac{2}{3}(6a+9)\) = 20.4

Using the distributive property, we get

\(\frac{2}{3} {6a+9}\) = 20.4

\(\frac{2}{3}\) ×  6a + \(\frac{2}{3}\)×9 = 20.4

2 × 2a + 2 × 3 = 20.4

4a + 6 = 20.4

4a + 6 = 20.4

4a = 20.4 − 6

4a = 14.4

a = \(\frac{14.4}{4}\)

a = 3.6

The value of a = 3.6

Given:

If you apply the Distributive property first to solve the equation, what operation will you need to use last?

To find/solve

Using the distributive property.

6 \(\frac{d}{3}\) − 30 = 34

​2d−30 = 34

2d = 64

d = 32
​
When a distributive property is used as the first step to solve the equation, the last operation will be division.

Given:

If instead, you divide first to solve the equation, what operation would you need to use last?

To find/solve

Using the distributive property.

d = \(\frac{192}{6}\)

d = 32

When a distributive property is used as the first step to solve the equation, the last operation will be division.

Envision Math Grade 7 Exercise 6.3 Solution Guide

Question. A local charity receives \(\frac{1}{3}\) of funds raised during a craft fair and a bake sale. The total amount given to charity was $137.45. How much did the bake sale raise?

Given:

A local charity receives \(\frac{1}{3}\) of funds raised during a craft fair and a bake sale.

The total amount given to charity was $137.45.

To find/solve

How much did the bake sale raise?

Determine the amount raised by the bake sale.

x = 159.75

The amount raised by the bake sale is $159.75.

Question. Diagram shown for the equation is incorrect. Find the correct solution.

Given that, the solution shown for the equation is incorrect.

We need to find the correct solution.

The given equation is  − 3(6 − r) = 6

Solving the given using the distributive property, we get

-3(6-r) = 6

-18 + 3r = 6

-18 + 18 + 3r = 6 + 18

3r = 24

r = \(\frac{24}{3}\)

r = 8

The value of  r = 8 not  r = −8

The multiplication of two negative integers result in a positive product.

Given that, the solution shown for the equation is incorrect.

We need to find the error made.

The given equation is − 3(6 − r) = 6

Solving the given using the distributive property, we get

​− 3(6 − r) = 6

− 18 + 3r = 6

3r = 24

r = \(\frac{24}{3}\)

r = 8

The multiplication of two negative integers results in a positive product. The multiplication of two negative integers results in a positive product. Here, he has multiplied two negative integers and put the result as negative. This was likely the error he made.

Question. Vita wants to center a towel bar on her door that is \(27\frac{1}{2}\) inches wide. She determines that the distance from each end of the towel bar to the end of the door is 9 inches.

Given:

Vita wants to center a towel bar on her door that is \(27\frac{1}{2}\)inches wide.

She determines that the distance from each end of the towel bar to the end of the door is 9 inches.

Let x inches be the length of the towel bar.

The width of the door is 9 inches on each side of the towel bar.

So the width of the door is

9 + x + 9 = x + 18inches

It’s given that the width of the door is \(27\frac{1}{2}\)

∴ \(27\frac{1}{2}\) = 18 +x

Now we solve the equation:

\(27\frac{1}{2}\) = 18 + x

\(27\frac{1}{2}\) − 18 = 18 + x − 18 ( Subtract −18 on both sides)

x = 9\(\frac{1}{2}\)

x = 4.5

Therefore, the length of the towel bar is x = 4.5

Envision Math Accelerated Grade 7 Chapter 6 Exercise 6.3 Answers

Question. Apply the properties of equality to isolate the variable.

We need to explain how to isolate the variable in the equation \(\frac{-2}{3}n+7\) = 15

Apply the properties of equality to isolate the variable.

\(\frac{-2}{3}n+7\)− 7 = 15−7

\(\frac{-2}{3}n\) = 8

\(\frac{-3}{2}n\)×\(\frac{-2}{3}n\)

=  8 × \(\frac{-2}{3}n\)

n =− 12

Use the properties of equality to isolate the variable in the equation to get the solution of
​
n =−12

Use the properties of equality to isolate the variable in the equation to get the solution of n =−12.

Question. Find the equations which are equivalent to \(\frac{1}{2}\)(4+8x) = 17.

We need to find the equations which are equivalent to

\(\frac{1}{2}\)(4+8x) = 17

Determine the equation similar to the given

\(\frac{4}{2}\)  + \(\frac{8}{2}\)x = 17

2 + 4x = 17

4x = 15

x = \(\frac{15}{4}\)

​Determine the equation similar to the given
​
\(\frac{4+8 x}{2}\) = 17
​
2.(\(\frac{4+8 x}{2}\)) = 17.2

​4 + 8x = 34

8x = 30

x =  \(\frac{30}{8}\)

​The similar equations are 4 + 8x = 34 and 4x = 15.

Question. Clara has 9 pounds of apples. She needs 11/4 pounds to make one apple pie. If she sets aside 1.5 pounds of apples. How many apple pies clara makes.

Given:

Clara has 9 pounds of apples. She needs 11/4 pounds to make one apple pie. If she sets aside 1.5 pounds of apples.

The equation for the given situation is 1\(\frac{1}{4}\)p = 9-1.5

1\(\frac{1}{4}\)p = 9 −1.5

\(\frac{5}{4p}\) = 7.5

5p = 4(7.5)

5p = 30

p = 6

Clara can make 6 apple pies.

Question. Solve the given equation -4(1.75 + x) = 18

Given:

− 4 (1.75 + x) = 18

we have to solve the given equation.

First, we will apply distributive property and apply other properties of equality.

Given equation is − 4(1.75 + x) = 18

7 − 4x = 18                   (Distributive property)

− 7− 4x + 7 = 18 + 7   (Adding 7 to both sides)

−4x = 25

\(\frac{-4x}{-4}\)=\(\frac{25}{-4}\)                  (Dividing both sides by− 4)

x = −6.25

The value of x is  − 6.25

Envision Math 7th Grade Exercise 6.3 Step-By-Step Solutions

Question. A group of 4 friends travel in a train ticket cost for one person $6 and total amount they spent by 4 friends is $34 they buy a healthy snack bag find the value of healthy snack bag. write the equation which represents the total amount spent and then we will solve it.

Given:

Ticket cost for one person = $6

Cost of healthy snack bag = $b

The total amount spent by 4 friends is $34

We have to find the value of b.

First, we will write the equation which represents the total amount spent and then we will solve it.

Given:

Ticket cost for one person = $6

Cost of healthy snack bag = $b

The total amount spent by 4 friends is $34

Therefore,

Amount spent by one friend = 6 + b

Total amount spent by 4 friends = 4(6 + b) = 34

4 (6 + b) = 34

24 + 4b = 34                   (Distributive property)

24 + 4b −24 = 34 − 24  (Subtracting 24 from both sides)

4b = 10

\(\frac{4b}{4}\)=\(\frac{10}{4}\)                       (Dividing both sides by 4)

b = 2.5

Hence, the cost of a healthy snack bag is $2.50

Equation is 4(6 + b) = 34

The cost of a healthy snack bag is $2.50.

Envision Math Accelerated Grade 7 Volume 1 Chapter 6 Solving Problems Using Equations And Inequalities

Question. Explain how the expression 2m + 3 relates to the values in the table.

Given:

Golf balls in Marley’s collection are m

Golf balls in Tucker’s collection are 2m + 3

We have to explain how the expression 2m + 3 relates to the values in the table.

The expression 2m+3 showed the comparison of the number of golf balls in Marley’s and Trucker’s collections.

Golf balls in Marley’s collection are m

Golf balls in Tucker’s collection are 2m + 3

From the expression 2m + 3, it is clear that it represents how greater the number of golf balls in Tucker’s collection are compared to Marley’s collection.

The expression 2m + 3 showed the comparison of a number of golf balls in Marley’s and Tucker’s collections.

Envision Math Accelerated Grade 7 Chapter 6 Exercise 6.1 Answer Key

Question. The total price of the laptop is $335. The down payment $50 and Cole pays the rest of the money in 6 equal monthly payments. Write an equation that represents the relationship between the cost of the laptop and Cole’s payments.

Given:

The total price of the laptop is $335.

The down payment $50 and Cole pays the rest of the money in 6 equal monthly payments ‘p’.

We have to write an equation that represents the relationship between the cost of the laptop and Cole’s payments.

The total price of the laptop is $335.

The down payment is $50 and Cole pays the rest of the money in 6 equal monthly payments ‘p’.

Cost  = (50) + 6 ×  monthly payments

335 = (50) + 6 × p

335 = 50 + 6p

Therefore, 335 = 50 + 6p represents the relationship between the cost of the laptop and Cole’s payments.

335 = 50 + 6p represents the relationship between the cost of the laptop and Cole’s payments.

Question. The total money Claire spend is $9.49. There is 60% off on the hat and socks price is $5.49. Explain which equation represents correctly Claire’s shopping trip.

Given:

The total money Claire spend is $9.49

There is 60% off on the hat and socks price is $5.49.

The original price of hat is $x.

We have to explain which equation represents correctly Claire’s shopping trip.

First, we will calculate the price she paid for the hat and then add it to the price of the socks, this will give us the total price she paid.

The original price of hat is $x.

There is 60% off on the hat.

Therefore, Claire paid only 40% of the original price of the hat.

Hence, the price she paid for hat 40%x  = b 0.4x

She paid $5.49 for socks and in total, she paid $9.49.

Therefore,  Hat’s price added to the socks price will give us the total amount she spent.

Hence, 0.4x +  5.49 = 9.49

0.4x  +  5.49   = 9.49 represents Claire’s shopping trip.

Envision Math Grade 7 Chapter 6 Solving Equations And Inequalities Exercise 6.1 Solutions

Question.  Explained why both multiplication and addition are used in Cole’s payment.

Given:  335 = 50 + 6p

We have to explain why both multiplication and addition used in the equation.

Cole’s payment equation 335 = 50 + 6p I

In this equation, we used both multiplication and addition because:

We used multiplication to get the total monthly payment Cole paid in 6 months

Therefore, we get, 6 × p = 6p

To get the total cost, we have to add a down payment which is $50 to the monthly payments Cole made.

Therefore, we got the equation 335=50+6p by using both multiplication and addition.

Explained why both multiplication and addition are used in Cole’s payment equation, which is 335 = 50 + 6p

We need to check whether the equations \(\frac{1}{5}x\) + 2 = 6 and \(\frac{1}{5}\)(x+2) = 6 represent the same situation.

Simplifying the equations, we get

\(\frac{1}{5}x\) + 2 = 6

\(\frac{1}{5}x\)  = 6 – 2

\(\frac{x}{5}\) = 4

x = 20

The other one will be

\(\frac{1}{5}\)(x+2) =  6

x + 2 = 30

x = 28

Thus, both represent different situations.

Both equations represent different situations.

Question. If Cora has 36 apps, and Rita started some apps and Deleted 5 apps determine the apps Rita has.

Given:

Rita started “r” apps

Deleted 5 apps

Cora has twice of Rita

The equation that represents the number of apps

​2c = r − 5

c =  \(\frac{r-5}{2}\)

r − 5 = 2c

r − 5 + 5 = 2c + 5

r = 2c + 5

If Cora has 36 apps, determine the apps Rita has.

​r = 2c + 5

r = 2(36) + 5

r = 72 + 5

r = 77

r = 2c + 5 represents the number of apps each girl has.

Question. The search collected 19 stamps. Her collection is 6 less than 5 times the collection of Jessica how many stamps are in her collection of Jessica?

Given, Her collection is 6 less than 5 times the collection of Jessica.

The search collected 19 stamps. Her collection is 6 less than 5 times the collection of Jessica how many stamps are in her collection of Jessica?

The problem is about the stamp collection of Jessica.

The collection of Sarah is 6 less than 5 times the collection of Jessica.

Given:

11 hours this week

5 fewer

The equation that represents the number of hours for her babysitting

\(\frac{2}{3}\)h-5 = 11

The equation is \(\frac{2}{3}\)h-5 = 11

Question. The relationship between the weight of the crate and the number of oranges.

Given:

Total weight 24.5 pounds. One orange 0.38 lb

The equation that represents the relationship between the weight of the crate and the number of oranges is

24.5 = 15 + 0.38 × g

The equation is 24.5 = 15 + 0.38 × g.

Given:

37 guests

3 large table

7 late arriving table

The equation that represents the situation is

3n + 7 = 37

The equation is  3n + 7 = 37.

Given:

Each friend paid $12.75

Tip was $61

The equation that represents the situation is  4(12.75 + t) = 61.

The equation is  4(12.75 + t) = 61.

Solving Problems Using Equations And Inequalities Grade 7 Exercise 6.1 Envision Math

Question. Mia buys \(4 \frac{1}{5}\) pounds of plums. The total cost after using a coupon for 55 off her entire purchase was $3.23. Find the equation could represent the situation.

Given that, Mia buys \(4 \frac{1}{5}\) pounds of plums. The. total cost after using a coupon for 55 ¢ off her entire purchase was $3.23.

If c represents the cost of the plums in dollars per pound, we need to find what equation could represent the situation

The equation that represents the situation is  \(4 \frac{1}{5}\)  c−0.55 = 3.23.

The equation is  \(4 \frac{1}{5}\) c−0.55 = 3.23.

The situation could still be used even if the denominator is doubled.

Even if the denominator will be doubled, the situation could still be used as for making a larger field and making more buses to be used in the field trip.

The situation could still be used even if the denominator is doubled.

Question. Iguana costs $48. You already have $12 and plan to save $9 per week. We have to form an equation that represents the plan to afford the iguana.

Iguana costs $48. You already have $12 and plan to save $9 per week. We have to form an equation that represents the plan to afford the iguana.

Let w represent the number of weeks.

We form an equation:

12 + 9w = 48

This is the required equation.

Therefore, the required equation that represents the plan to afford the iguana is  12 + 9w = 48

Given:

Iguana costs $48. You already have $12 and you can buy the iguana after 6 weeks.

We have to form an equation that represents the amount that needs to be saved per week.

Let x represent the amount to be saved per week.

We form an equation:

12 + 6x = 48

This is the required equation.

Therefore, the required equation that represents the amount to save per week to buy the iguana is  12 + 6x = 48

Envision Math Accelerated Grade 7 Chapter 6 Exercise 6.1 Answers

Question. The life expectancy of a woman born in 1995 was 80.2 years. Find the equation for the life expectancy of a woman born in 2005.

The life expectancy of a woman born in 1995 was 80.2 years.

Between 1995 and 2005, the life expectancy increased by 0.4 years every 5 years.

We have to find the equation for the life expectancy of a woman born in 2005.

Let L represent the life expectancy of a woman born in 2005.

We are given that the life expectancy increased by 0.4 every 5 years.

Therefore, the life expectancy will increase by 0.8 in 2005.

We form equations:
​
​L − 0.4(2) = 80.2

L − 80.2 = 0.4(2)

20.2 + 0.4(2) = L
​
These are the required equations.

Therefore, the required equations that represent the life expectancy of a woman born in 2005 are, ​L − 0.4(2) = 80.2, L − 80.2 = 0.4(2), 20.2 + 0.4(2) = L

The equation that we formed for this particular problem is a linear equation.

The equations should be equivalent so that we get the same answer.

The equations can look different but should be rewritten as  L = 80.2 + 0.4(2)

No, the equations must be equivalent, they can look different but should be able to rewrite as L = 80.2 + 0.4(2)

Question. Solve the equation 5x – 13 = 12 and multiply x 5 times.

We are given the equation  5x−13 = 12

This equation shows that we have to multiply x 5 times.

Then deduct 13 from 5x, so that we will get the answer as 12.

We need an equation such that 5 times the value of x minus  13 is 12.

Given:

5x−13=12

We put the values 1, 2, 3, 4, and 5 for x in the equation:

For 1:

​5(1) − 13 = 12

5 − 13 = 12
​
−8 ≠ 12

For 2:

​5(2) − 13 = 12

10 − 13 = 12
​
−3 ≠ 12

For 3:

​5(3) − 13 = 12

15−13 = 12
​
2 ≠ 12

For 4:

​5(4) − 13 = 12

20 − 13 = 12
​
7 ≠ 12

For 5:

​5(5) − 13 = 12

25 − 13 = 12
​
12 = 12

Therefore, the solution of the given equation is 5.

Envision Math Grade 7 Exercise 6.1 Solution Guide

Question.  A garden contains 135 flowers each of which is either red or yellow. These are 3 beds of yellow flowers and 3 beds of red flowers. There are 30 yellow flowers in each yellow flower bed. Find out number of flowers in each red flower bed.

A garden contains 135 flowers each of which is either red or yellow. There are 3 beds of yellow flowers and 3 beds of red flowers.

There are 30 yellow flowers in each yellow flower bed.

Let

z − total number of flowers in the garden=135

y − number of flowers in each yellow flower bed=30

r − Number of flowers in each red flower bed.

We form the equation:

z = 3y + 3r

​∴ 135 = 3(30) + 3r

135 = 90 + 3r

3r = 135 − 90

3r = 45

r = \(\frac{45}{3}\)

r = 15

The number of flowers in each red flower bed is 15.

Therefore, the equation that represents the number of red and yellow flowers is  135  =  90 + 3r

A garden contains 135 flowers, each of which is either red or yellow.

There are 3 beds of yellow flowers and 3 beds of red flowers.

There are 30 yellow flowers each yellow flower bed.

To find/solve:

Write another real-world situation that your equation from Part A could represent.

An equation is a statement of equality between two expressions consisting of variables and numbers.

The equation that represents the number of red and yellow flowers is  3r + 90  = 135.

The equation that represents the number of red and yellow flowers is   3r + 90  = 135.

Envision Math Accelerated Grade 7 Volume 1 Chapter 6 Solving Problems Using Equations And Inequalities

Question. How can you solve real-world and mathematical problems with numerical and algebraic equations and inequalities?

Given

Statement

To find/solve

How can you solve real-world and mathematical problems with numerical and algebraic equations and inequalities?

The equation shows the relationship between variables and other quantities in a situation which involves different operations on numbers.

The left side of an equation must have the same value with the right side to make the equality true.

The variable will also indicate balance in the relationship of the equation.

The left side of an equation must have the same value with the right side to make the equality true.

The left side of an equation must have the same value with the right side to make the equality true.

We need to research the need for safe, clean water in developing countries.

Based on the research, we need to determine the type, size, and cost of a water filtration system needed to provide clean, safe water to a community.

Also, we have to develop a plan to raise money to purchase the needed filtration system.

Reverse osmosis systems are the most effective filters for drinking water.

Many of them feature seven or more filtration stages along with the osmosis process that makes them.

Effective at moving 99 percent of contaminants from water, including chemicals such as chlorine, heavy metals, pesticides, and herbicides.

Filtered water reduces corrosion and improves PH levels also extending the life of household fixtures.

It costs around $30 per square foot.

We can organize community parties to fetch donations for raising money for the purchase of the needed filtration system.

Reverse osmosis systems are the most effective filters for drinking water. It helps us in getting safe, clean water in developing countries.

A property of equality states that performing the same operation on both sides of an equation will keep the equation true.

Properties that state that performing the same operation on both sides of an equation will keep the equation true are called properties of equality.

The inverse relationship is involved in addition and subtraction because they can “undo” each other.

Addition and subtraction have a(n) inverse relationship because they can “undo” each other.

Like terms are terms that have same the variables and powers

Example:

5x+10x

Terms that have the same variable are called like terms.

Envision Math Accelerated Grade 7 Chapter 6 Exercise Key Answers

Question. Find the value of x to solve the given equation x + 9.8 = 1.2

We need to use properties to solve the given equation for x

The given equation is  x + 9.8 = 14.2

The given equation is   x + 9.8 = 14.2

Solving it, we get

​x + 9.8 = 14.2

x + 9.8 − 9.8 = 14.2 − 9.8

x = 4.4

The value of x=4.4

Question. Find the value of x solve the given equation 14x = 91.

We need to use properties to solve the given equation for x

The given equation is  14x = 91

The given equation is 14x = 91

Solving it, we get

14x = 91

\(\frac{14x}{14}\)=\(\frac{91}{14}\)

x = 13

The value of x = 13

Question. Find the value of x solve the given equation \(\frac{1}{3}x\)

We need to use properties to solve the given equation for x

The given equation is  \(\frac{1}{3}x\) = 24

The given equation is  \(\frac{1}{3}\)x = 24

Solving it, we get

\(\frac{1}{3}\)x = 24

3×\(\frac{1}{3}\)x = 3 × 24

x = 3 × 24

x = 72
​
The value of x = 72

Envision Math Grade 7 Chapter 6 Solving Equations And Inequalities Solutions

Question. Solve the expression and combine like terms \(\frac{1}{4}+\frac{1}{4}m-\frac{2}{3}k+\frac{5}{9}m\).

We need to combine like terms in the given expression.

The given expression is \(\frac{1}{4} k+\frac{1}{4} m-\frac{2}{3} k+\frac{5}{9} m\)

The given is, \(\frac{1}{4} k+\frac{1}{4} m-\frac{2}{3} k+\frac{5}{9} m\)

Combining the like terms, we get

=   \(\frac{1}{4} k−\frac{2}{3} k + \frac{1}{4} m+\frac{5}{9} m\)

=   k(\(\frac{1}{4}\)−\(\frac{2}{3}\)) + m( \(\frac{1}{4}\) + \(\frac{5}{9}\))

=  k(\(\frac{3−8}{12}\)) + m(\(\frac{9+20}{36}\))

=  k (\(\frac{−5}{12}\)) + m (\(\frac{29}{36}\))

=\(\frac{−5}{12}\) k + \(\frac{29}{36}\)m

The expression becomes  \(\frac{-5}{12} k+\frac{29}{36} m\)

Solving Problems Using Equations And Inequalities Grade 7 Envision Math

Question. Solve the expression and combine like terms -4b + 2w + (-4b) + 8w.

We need to combine like terms in the given expression.

The given expression is −4b + 2w + (−4b) + 8w

The given is, −4b + 2w + (−4b) + 8w

Combining the like terms, we get

​−4b + 2w + (−4b) + 8w

= −4b + 2w − 4b + 8w

=  b(−4−4) + w(2 + 8)

= −8b + 10w

The expression becomes −8b + 10w

Question. Solve the expression and combine like terms 6 – 5z + 8 – 4z + 1.

We need to combine like terms in the given expression.

The given expression is 6−5z + 8 − 4z + 1

The given is 6 − 5z + 8−4z + 1

Combining the like terms, we get

​6−5z + 8 − 4z + 1

=  6 + 8 + 1 − 5z − 4z

=  15 − z(5 + 4)

= 15 − 9z

The expression becomes 15 − 9z

Envision Math Grade 7 Chapter 6 Exercise Key Solutions

Question. A large box of golf balls has more than 12 balls. We need to describe how your inequality represents the situation.

We need to write an inequality that represents the situation

A large box of golf balls has more than 12 balls.

We need to describe how your inequality represents the situation.

Given that a large box of golf balls has more than 12 balls.

Let x be the number of golf balls in a large box.

If it is more than that, it is represented by the symbol <

Thus, the inequality equation will be

​12 < x

x > 12
​
The inequality that represents the situation x > 12

Envision Math Accelerated Grade 7 Chapter 6 Exercise Answers

Question. Write the similarities and differences between an equation and an inequality.

We have to write the similarities and differences between an equation and an inequality.

Similarities between an equation and an inequality:

Equation and inequality both use variables when writing and expressing.

Just like the equations, the solution to equality is a value that makes the inequality true.

Both expressions may have different solutions.

Differences:

An equation is a mathematical statement that shows the equal value of two expressions.

While an inequality is a mathematical statement that shows that an expression is lesser than or more than the other.

An equation has only one definite value of a variable while an inequality may have several values because of its range.

Both equations and inequalities show the relation between two expressions. The equation shows that the two expressions are equal while inequalities show that an expression is bigger or smaller than the other.

Envision Math Accelerated Grade 7 Volume 1 Chapter 5 Generate Equivalent Expressions

Question. Explain whether sorting the tiles with positive coefficients together and tiles with negative coefficients.

We need to explain whether sorting the tiles with positive coefficients together and tiles with negative coefficients.

Together help to simplify an expression that involves all the tiles.

The way of sorting the tiles with positive coefficients together and tiles with negative coefficients together will not help much.

Since we only add all the numbers alone and subtract the rest from them.

The main problem here exists with the numbers with variables.

We couldn’t able to do any arithmetic operations on them.

So, this won’t help much.

Sorting the tiles with positive coefficients together and tiles with negative coefficients together help to simplify an expression. That involves all the tiles wouldn’t help much.

Question. Explain how the properties of operations are used to simplify expressions.

We need to explain how the properties of operations are used to simplify expressions.

Simplification is the process of rewriting the given expression into its most compact form.

To simplify expressions, we can use the properties of operations such as distributive, commutative, and associative properties.

These properties will help in simplifying the terms by grouping them together.

Also, it combines like terms to get the simplified form.

The properties of operations are used to simplify expressions by grouping and combining the like terms together.

Envision Math Accelerated Grade 7 Chapter 5 Exercise 5.3 Answer Key

Question. Simplify the expression -6 – 6f + 7 – 3f – 9.

We need to simplify the expression − 6 − 6f + 7 − 3f − 9

The given expression is  −6−6f  +7 − 3f − 9

Simplifying it we get

​−6−6f + 7 − 3f − 9 = − 6f − 3f − 6 + 7 − 9

= −9f + 1 − 9

= − 9f − 8
​
The simplified expression is  −9f − 8

Simplification is the process of rewriting the given expression into its most compact form.

To simplify expressions, we can use the properties of operations such as distributive, commutative, and associative properties.

These properties will help in simplifying the terms by grouping them together.

Also, it combines like terms to get the simplified form.

We can sort the expressions into two groups.

One having the numbers alone.

The other having both the numbers and variables in it.

We can group each group separately and then do arithmetic operations to simplify it.

We will decide in what way to reorder the terms of an expression. When simplifying it is by looking at the variables and constants and grouping them separately to simplify it.

Question. Explain how the properties of operations are used to simplify expressions.

We need to explain how the properties of operations are used to simplify expressions.

Simplification is the process of rewriting the given expression into its most compact form.

To simplify expressions, we can use the properties of operations such as distributive, commutative, and associative properties.

These properties will help in simplifying the terms by grouping them together.

Also, it combines like terms to get the simplified form

The properties of operations are used to simplify expressions by grouping and combining the like terms together

Envision Math Grade 7 Chapter 5 Equivalent Expressions Exercise 5.3 Solutions

Question. Explain why constant terms expressed as different rational number types can be combined.

We need to explain why constant terms expressed as different rational number types can be combined.

The constant terms expressed as different rational number types can be combined because can be grouped together since they are all like terms.

The constant terms don’t have a variable in them.

So it can be grouped. It can be simplified, grouped, and combined no matter what since they are all like terms.

Constant terms expressed as different rational number types can be combined because they are all like terms.

Question. Explain how you know when an expression is in its simplest form.

We need to explain how you know when an expression is in its simplest form.

The expression is in its simplest form when it cannot be simplified further.

We cannot combine or group any further terms if they are simplified.

We cannot use any properties to simplify it further when they are simplified.

An expression is in its simplest form when we cannot group or combine them further.

Question. Simplify the expression -4b + (-9k) – 6 – 3b + 12.

We need to simplify the expression  −4b + ( −9k)−6 − 3b + 12.

The given expression is  −4b + (−9k) −6 −3b + 12

Simplifying it we get

​−4b + (−9k) − 6 − 3b + 12 = −4b − 9k − 6 − 3b + 12

= −4b − 3b − 9k − 6 + 12

= −7b − 9k + 6
​
The simplified expression is −7b − 9k + 6

Question. Simplify the expression -2 + 6.45z – 6 + (-3.25z).

We need to simplify the expression  −2 + 6.45z − 6 + ( −3.25z)

The given expression is  −2 + 6.45z − 6 + (−3.25z)

Simplifying it we get

​−2 + 6.45z − 6 + (−3.25z) =−2 − 6 + 6.45z − 3.25z

= −8 + 3.20z

=  3.2z − 8

The simplified expression is 3.2z − 8

Question. Simplify the expression -9 + (\(\frac{-1}{3}y\)) + 6-\(\frac{4}{3}y\).

We need to simplify the expression

− 9 + (\(\frac {-1}{3}y\)) + 6−\(\frac {4}{3}y\)

The given expression is − 9 + (\(\frac {-1}{3}y\)) + 6 −\(\frac {4}{3}y\)

Simplifying it, we get

− 9  +  (\(\frac{−1}{3}y\))+6−\(\frac{4}{3}y\)

= − 9 + 6 − \(\frac{1}{3}y\) − \(\frac{4}{3}y\)

= −3 + (\(\frac{ −y −4y}{3}\))

= −3 −\(\frac{5}{3}y\)

The simplified expression is − 3 −\(\frac {5}{3}y\)

Generate Equivalent Expressions Grade 7 Exercise 5.3 Envision Math

Question. Simplify the expression -2.8f + 0.9f – 12 – 4.

We need to simplify the expression − 2.8f + 0.9f − 12 − 4

The given expression is − 2.8f + 0.9f − 12 − 4

Simplifying it, we get

​− 2.8f + 0.9f − 12 − 4 = f(−2.8 + 0.9) − 16

= f( −1.9) −16

= −1.9f − 16
​
The simplified expression is −1.9f − 16

Question. Simplify the expression 3.2 – 5.1n – 3n + 5.

We need to simplify the expression  3.2 − 5.1n − 3n + 5

The given expression is  3.2 − 5.1n − 3n + 5

Simplifying it we get

​3.2 − 5.1n − 3n + 5

=  3.2 + 5 − 5.1n − 3n

=  8.2 − 8.1n
​
The simplified expression is  8.2 − 8.1n

Question. Simplify the given expression 2n + 5.5 – 0.9n – 8 + 4.5p.

We need to simplify the given expression.

The given expression is 2n + 5.5 − 0.9n − 8 + 4.5p

Combining the like terms together, we get

​2n + 5.5 − 0.9n − 8 + 4.5p

=  2n − 0.9n + 5.5 − 8 + 4.5p

=  n(2 − 0.9) − 2.5 + 4.5p

=  1.1n + 4.5p − 2.5

The simplified expression is 1.1n + 4.5p − 2.5

Question. Simplify the given expression 12 + (-4)\(-\frac{2}{5}j-\frac{4}{5}j\) + 5.

We need to simplify the given expression.

12 + (−4)\(−\frac{2}{5} j−\frac{4}{5} j\) + 5

Using the properties and combining the like terms together, we get

12 +  (−4)\(-\frac{2}{5} j-\frac{4}{5} j\) + 5

=   12− 4\(-\frac{2}{5} j−\frac{4}{5} j\) + 5

=   8 +  j(\(-\frac{2}{5} −\frac{4}{5} \)) + 5

=  13 +  j(\(\frac{−2−4}{5}\))

=  13 +  j\(\frac{−6}{5}\)

=  13− j\(\frac{6}{5}\)

The simplified expression is 13−\(\frac{6}{5} j\)

Envision Math Grade 7 Exercise 5.3 Solution Guide

Question. Simplify the given expression -5v + (-2) + 1 + (-2v) and to find which among the given expression is equal to it.

We need to simplify the given expression − 5v + ( −2) + 1 + (−2v) and to find which among the given expression is equal to it

Simplifying it, we get

​−5v + (−2) + 1 + ( −2v) = −5v − 2 + 1 − 2v

= −5v − 2v − 2 + 1

= −7v − 1

The given expression is equal to (C) −7v − 1

Question. Find which expression is equivalent to \(\frac{2}{3}x+(-3)+(-2)-\frac{1}{3}x\).

We need to find which expression is equivalent to \(\frac{2}{3}x+(-3)+(-2)-\frac{1}{3}x\)

Simplifying it, we get

\(\frac{2}{3}\)x +(−3)+(−2)−\(\frac{1}{3}\)x

= \(\frac{2}{3}\)x −3−2−\(\frac{1}{3}\)x

=  \(\frac{2}{3}\)x − \(\frac{1}{3}\)x −5

=  \(\frac{2x −x}{3}\) − 5

=  \(\frac{x}{3}\) − 5

=  (or)  \(\frac{1}{3}x\) − 5

The given expression is equivalent to  \(\frac{1}{3}x\) − 5

Question. The dimensions of a garden are shown. We need to write an expression to find the perimeter.

The dimensions of a garden are shown. We need to write an expression to find the perimeter.

Add all the lengths to find the perimeter.

The perimeter of the rectangle is 2(l+w)

Here, substituting the given we get

2(1+w) = 2(\(\frac{1}{2}x\) −7+x)

=  2(\(\frac{1}{2}x\) +x−7)

= 2 (\(\frac{x+2x}{2}−7\))

=  2(\(\frac{3x}{2}\)−7)

=  3x−14

The expression to find the perimeter is 3x−14

Question. Simplify the given expression 8h + (-7.3d) – 14 + 5d – 3.2h

We need to simplify the given expression.

The given expression is  8h +(−7.3d) −14 + 5d − 3.2h

Combining the like terms together, we get

​8h + (−7.3d) − 14 + 5d − 3.2h

=  8h −7.3d −14 + 5d − 3.2h

=  8h − 3.2h − 7.3d + 5d − 14

=  4.8h − 2.3d − 14

The simplified expression is  4.8h − 2.3d − 14

Question. Simplify the given expression 4 – 2y + (-8y) + 6.2.

We need to simplify the given expression.

The given expression is  4 − 2y + (−8y) + 6.2

Combining the like terms together, we get

​4−2y + (−8y) + 6.2 = 4 − 2y − 8y + 6.2

=  4 − 10y + 6.2

=  4 + 6.2 − 10y

=  10.2 − 10y

The simplified expression is 10.2 − 10y

Envision Math Accelerated Grade 7 Chapter 5 Exercise 5.3 Answers

Question. Simplify the given expression \(\frac{4}{9}z-\frac{3}{9}z+5-\frac{5}{9}z-8\).

We need to simplify the given expression.

The given expression is \(\frac{4}{9} z-\frac{3}{9} z+5-\frac{5}{9} z-8\)

Combining the like terms together, we get

\(\frac{4}{9}z\)−\(\frac{3}{9}z\) + 5 −\(\frac{5}{9}z\) − 8

= \(\frac{4z−3z}{9}\)+5−\(\frac{5}{9}z\) − 8

= \(\frac{z}{9}\)−\(\frac{5z}{9}\) + 5 − 8

= \(\frac{−4}{9}z\) − 3 

The simplified expression is \(\frac{− 4z}{9} − 3\)

Question. Show that the given two expressions 11t – 4t is equivalent to 4t – 11t.

We need to explain whether  11t − 4t  is equivalent to  4t −11t.

We need to support our answer by evaluating the expression for t = 2.

The expression  11t − 4t  is not equivalent to  4t −11t.

Substitute t = 2 in both cases, we get

​11t −4t = 7t

=  7 × 2 = 14

4t−11t  = − 7t

= − 7 × 2 = −14

Thus, the value differs.

The given two expressions are not equivalent to each other.

Envision Math 7th Grade Exercise 5.3 Step-By-Step Solutions

Question. Observe the diagram the signs show the costs of different games at a math festival. Find how much would it cost n people to play Decimal Decisions and Ratio Rage.

Given that the signs show the costs of different games at a math festival.

We need to find how much would it cost n people to play Decimal Decisions and Ratio Rage.

The cost of playing Decimal decisions will be

12.70 − n + 9

The cost of Ratio Race will be

The total cost will be

12.70  −  n  + 9 +  \(\frac{n}{4}\)

=  12.7 −  n  + \(\frac{n}{4}\) + 9

= 12.7  + 9 + \(\frac{−4n+n}{4}\)

=  21.7 − \(\frac{3n}{4}\)

The cost to play Decimal Decisions-and Ratio Rage will be  21.7−\(\frac{3n}{4}\)