## 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.

**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.

**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.

**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

**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.

**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.