Chapter 4 Represent And Solve Equations And Inequalities
Section 4.1: Understand Equations And Solutions
Page 177 Exercise 1 Answer
Write an equation where the left side represents adding 10 cubes to the pan with 3 cubes and adding 4 cubes to the pan with 9 cube. It the sides are equal, that is if the equation is true, the pans will balance.
10 + 3 = 4 + 9
13 = 13
The pans would balance.
Result
The pans would balance and the equation is 10 + 3 = 4 + 9.
Read And Learn More: enVisionmath 2.0 Grade 6 Volume 1 Solutions
Page 177 Exercise 1 Answer
To make the pans balance write an equation, let’s say the left side has 3 cube and the right side has 9 cubes.
One way to make the equation true is to add 6 additional cubes to the left side.
3 + 6 = 9
There are 12 cubes in all on the balance so a second way to make the equation true is to remove 3 cubes from the right side and add them tot he right side since this would give 6 cubes on each side.
3 + 3 = 9 – 3
Result
One way is to add 6 additional cubes to the left side. A second way is to remove 3 cubes from the right side and add them to the left side.
Page 178 Exercise 1 Answer
To determine which app Tracy should buy substitute the variable x with its possible values, that is prices of apps and check for which value of x the equation
$21.00 = x + $9.00
is true.
For x = $9.50.
$21.00 = x + $9.00
= $9.50 + $9.00
= $18.50 ≠ $21.00
For x = $10.50.
$21.00 = x + $9.00
= $10.50 + $9.00
= $19.50 ≠ $21.00
For x = $12.00.
$21.00 = x + $9.00
= $12.00 + $9.00
= $21.00
The equation is true for x = $12.00. Thus, Tracy bought a remote desktop.
Three values are possible, however two of them, $9.50 and $10.50, are not whole numbers, but $12.00 is.
The sum of a whole number and a decimal number will be a decimal number. Since on the left side is a whole number, and to x we add $9.00 – a whole number, x must as well be a whole number .
Thus, we know that the solution is x = $12.00, without using substitution.
Result
x = $12.00, Remote Desktop.
Page 179 Exercise 2 Answer
Use substitution – that is substitute y for each of the values and evaluate to see if any of Anthony’s friends guessed correctly.
y = 48
y ÷ 4 = 48 ÷ 4
=12 ≠ 13
y = 60
y ÷ 4 = 60 ÷ 4
= 15 ≠ 13
y = 120
y ÷ 4 = 120 ÷ 4
= 30 ≠ 13
None of Anthony’s friends guessed correctly.
Step 2
To find how many marbles Anthony has, multiply both sides of the equation by four.
y ÷ 4 = 13
y = 13 × 4
y = 52
Anthony has 52 marbles in all.
Result
None of Anthony’s friends guessed correctly, he has 52 marbles in all.
Page 180 Exercise 1 Answer
To determine whether a given number makes an equation true use substitution. Replace the variable with its value and evaluate both sides of the equation to make sure they have the same value. If they do, the given number makes the equation true.
Page 180 Exercise 2 Answer
An equation is true when both the left and the right side of the equation have the same value.
That is, if the equation contains a variable or variables, it is true for the variable or variables for which both sides have equal values.
For example, the equation
x + 15 = 18,
is true for x = 3, since
3 + 15 = 18.
However it is not true for x = 4, since
4 + 15 = 19 ≠ 18.
Result
An equation is true when both the left and the right side of the equation have the same value.
Page 180 Exercise 3 Answer
To check whether Ben is correct substitute the variable n in the expression 7n = 45 vith its value, that is with 5, as Ben says.
7n = 45
7(5) = 45
35 ≠ 45
Ben is not correct, n = 5 is not the solution of the equation.
However, n = 9 is the solution, since
7n = 45
7(9) = 45
45 = 45.
Result
Ben is not correct and n = 5 is not the solution of the equation.
Page 180 Exercise 4 Answer
To find for which c the equation 3 + c = 11 is true, use substitution and check Lucy’s answers, that is substitute c with 7, 8, 9, and 10, or do the following:
3 + c = 11
c = 11 – 3 (Subtract 3 from both sides.)
c = 8
Lucy should add 8 cubes to make the pans balance.
Result
8
Page 180 Exercise 5 Answer
The equation is d + 9 = 35.
The given values of d are 16, 22, 26, and 36. We will substitute d with a value to check whether any of them are the solution.
16 + 9 = 25 ≠ 35
22 + 9 = 31 ≠ 35
26 + 9 = 35
36 + 9 = 45 ≠ 35
The solution of the equation d + 9 = 35 is d = 26.
Result
d = 26
Page 180 Exercise 6 Answer
The equation is 14n = 35.
The given values of n are 2, 3, 3.5, and 4. We will substitute n with a value to check whether any of them are the solution.
14 ⋅ 2 = 28 ≠ 35
14 ⋅ 3 = 42 ≠ 35
14 ⋅ 3.5 = 49 ≠ 35
14 ⋅ 4 = 56 ≠ 35
Result
None of the given values are the solution of the equation 14n = 35.
Page 180 Exercise 7 Answer
The equation is 13.4 − g = 8.1.
The given values of g are 4.3, 5.3, 5.5, and 6.5. We will substitute g with a value to check whether any of them are the solution.
13.4 – 4.3 = 9.1 ≠ 8.1
13.4 – 5.3 = 8.1
13.4 − 5.5 = 7.9 ≠ 8.1
13.4 − 6.5 = 6.9 ≠ 8.1
The solution of the equation 13.4 − g = 8.1 is g = 5.3.
Result
g = 5.3
Page 180 Exercise 8 Answer
The equation is 4 = 36 ÷ m.
The given values of m are 4, 6, 8, and 9. We will substitute m with a value to check whether any of them are the solution.
36 ÷ 4 = 9 ≠ 4
36 ÷ 6 = 6 ≠ 4
36 ÷ 8 = \(\frac{9}{4}\) = \(2 \frac{1}{4}\) ≠ 4
36 ÷ 9 = 4
The solution of the equation 4 = 36 ÷ m is m = 9.
Result
m = 9
Page 180 Exercise 9 Answer
The equation is n = 54 − 36.
We will substitute n with 8 to check whether the equation is true for n = 8.
54 − 36 = 18 ≠ 8
Result
The equation is not true for n = 8.
Page 180 Exercise 10 Answer
The equation is 5n = 40.
We will substitute n with 8 to check whether the equation is true for n = 8.
5 ⋅ 8 = 40
Result
The equation is true for n = 8.
Page 180 Exercise 11 Answer
The equation is 152 ÷ n = 21.
We will substitute n with 8 to check whether the equation is true for n = 8.
152 ÷ 8 = 19 ≠ 21
Result
The equation is not true for n = 8.
Page 180 Exercise 12 Answer
The equation is n + 46 = 54.
We will substitute n with 8 to check whether the equation is true for n = 8.
8 + 46 = 54
Result
The equation is true for n = 8.
Page 181 Exercise 13 Answer
The equation is t − 2.1 = 0.
The given values of t are 2.1, 2.4, 2.6, and 2.8. We will substitute t with a value to check whether any of them are the solution.
2.1 – 2.1 = 0
2.4 – 2.1 = 0.3 ≠ 0
2.6 – 2.1 = 0.5 ≠ 0
2.8 − 2.1 = 0.7 ≠ 0
The solution of the equation t − 2.1 = 0 is t = 2.1.
Result
t = 2.1
Page 181 Exercise 14 Answer
We are given the equation 49 = 7r and need to determine if r = 3, 6, 7, or 9 is a solution.
Substitute each value of r into the right side and then simplify to see if it equals 49:
r = 3 : 7r = 7(3) = 21 ≠ 49
r = 6 : 7r = 7(6) = 42 ≠ 49
r = 7 : 7r = 7(7) = 49 ✓
r = 9 : 7r = 7(9) = 63 ≠ 49
The solution of the equation is then r = 7.
Result
r = 7
Page 181 Exercise 15 Answer
The equation $4.10 = $6.25 – y is given. Four possible solution are offered, to tell which is true use substitution. Substitute the variable y with each value and evaluate the expression to check whether it is true or false.
For y = $2.15
$4.10 = $6.25 – y
= $6.25 – $2.15
= $4.10
For y = $2.95
$4.10 = $6.25 – y
= $6.25 – $2.95
≠ $3.30
For y = $3.05.
$4.10 = $6.25 – y
= $6.25 – $3.15
≠ $3.20
For y = $3.15
$4.10 = $6.25
= $6.25 – $3.15
≠ $3.10
Result
y = $2.15
Page 181 Exercise 16 Answer
The equation 24 ÷ h = 6 is given. Four possible solution are offered, to tell which is true use substitution. Substitute the variable h with each value and evaluate the expression to check whether it is true or false.
For h = 1.
24 ÷ h = 6
24 ÷ 1 = 6
24 ≠ 6
For h = 3.
24 ÷ h = 6
24 ÷ 3 = 6
8 ≠ 6
For h = 6.
24 ÷ h = 6
24 ÷ 6 = 6
4 ≠ 6
For h = 8.
24 ÷ h = 6
24 ÷ 8 = 6
3 ≠ 6
Result
None of the offered solutions are correct.
Page 181 Exercise 17 Answer
Miles per hour is a measurement for speed, as it states clearly, it tells how many miles one covers in an hour.
Marcie’s father rode his bike 108 miles in 7.5 hours, let’s calculate how fast he was going. Divide 108 by 7.5 since that is how much miles he covered in the time he spent cycling.
108 ÷ 7.5 = 14.4
Macie’s father rode his bike 14.4 mph.
y marks the number of hours she spent cycling. Use the equation 108 ÷ y = 18 to find how long it took her to cycle 108 miles if she rode her bike at the speed of 18 mph.
108 ÷ y = 18
108 = 18y (Multiply both sides b y.)
108 ÷ 18 = y (Divide both sides by 18.)
y = 6
It took Marcie 6 hours to cycle 108 miles. She didn’t match either of her parents time.
Result
y = 6
Page 181 Exercise 18a Answer
To check if b = 6 is the solution use substitution. Substitute the variable b for its value in the given equation and evaluate.
8b = 48
8(6) = 48
48 = 48
Result
b = 6 is the solution.
Page 181 Exercise 18b Answer
To check if b = 6 is the solution use substitution. Substitute the variable b for its value in the given equation and evaluate.
11 – b = 6
11 – 6 = 6
5 ≠ 6
Result
b = 6 is not a solution.
Page 181 Exercise 18c Answer
To check if b = 6 is the solution use substitution. Substitute the variable b for its value in the given equation and evaluate.
b + 3 = 9
6 + 3 = 9
9 = 9
Result
b = 6 is the solution.
Page 181 Exercise 18d Answer
To check if b = 6 is the solution use substitution. Substitute the variable b for its value in the given equation and evaluate.
54 ÷ b = 9
54 ÷ 6 = 9
9 = 9
Result
b = 6 is the solution.
Page 181 Exercise 19 Answer
To find the answer, that is which activity should the friends choose subtitute n with 4, since it is a group of 4 friends, in each of the equations. The equation which is true for n = 4 is the one tied to the activity they should choose.
6n + 5 = 29
6(4) + 5 = 29
24 + 5 = 29 (Substitute n with 4.)
29 = 29 (Find the product.)
14n = 29
14(4) = 29 (Substitute n with 4.)
56 ≠ 29 (Find the sum.)
30n – 40 = 29
30(4) – 40 = 29 (Substitute n with 4.)
120 – 40 = 29 (Find the product.)
80 ≠ 29 (Find the difference.)
The friends should choose the activity tied to the first equation, that is a Raft Trip.
Result
The friends should choose the Raft Trip.
Page 182 Exercise 20 Answer
To find which one of them is right use substitution – substitute p for each of the values they give as solutions.
27 = 18 + p
27 = 18 + 5 (Substitute p with 5.)
27 ≠ 23 (Evaluate and compare.)
27 = 18 + p
27 = 18 + 8 (Substitute p with 5.)
27 ≠ 26 (Evaluate and compare.)
27 = 18 + p
27 = 18 + 9 (Substitute p with 5.)
27 = 27 (Evaluate and compare.)
Rachel is correct – if 9 pennies are added the pans will balance.
Result
Rachel is correct.
Page 182 Exercise 21 Answer
Since Gerard spend $5.12 in total, that is for a sandwich and a drink, that is one side of the equation. Since s stands for the price of the sandwich and the drink cost $1.30, their sum is the other side of the equation.
s + $1.30 = $5.12
Substitute for the price of each sandwich to find for which one the equation is true. That is the sandwich Gerard bought.
s + $1.30 = $5.12
$3.54 + $1.30 = $5.12 (substitute s for 3.54)
$4.84 ≠ $5.12
s + $1.30 = $5.12
$3.82 + $1.30 = $5.12 (substitute s for 3.82)
s + $1.30 = $5.12
$3.92 + $1.30 = $5.12 (substitute s for 3.92)
$4.22 ≠ $5.12
Gerard bought a tuna sandwich for 3.82
Result
Gerard bought a tuna sandwich for 3.82.
Page 182 Exercise 22 Answer
x will be the variable in the equation. Since 12 is the solution the equation must be true for x = 12.
83 = 83 (choose any number and write it on both sides)
12 + 71 = 83 (Write 83 as a sum. Don’t change the other side)
12 + 1 + 70 = 83 (Write 71 as a sum)
12 + 1 + 7 x 10 = 83 (Write 70 as a product)
x + 1 + 7 x 10 = 83 (substitute 12 with x)
We used substitution but the other way round. Instead of subsituing a variable by its value, we decided that the value will be 12 and than wrote an expression which is true and put x instead of 12.
Result
x + 1 + 7 x 10 = 83
Page 182 Exercise 23 Answer
Substitute m for each of the values and evaluate to find for which value the equation is true.
255 – m = 124
255 – 111 = 124 (Substitute m with 111.)
144 ≠ 124 (Evaluate and compare.)
255 – m = 124
255 – 121 = 124 (Substitute m with 121.)
134 ≠ 124 (Evaluate and compare.)
255 – m = 124
255 – 131 = 124 (Substitute m with 131.)
124 = 124 (Evaluate and compare.)
255 – m = 124
255 – 141 = 124(Substitute m with 141.)
114 ≠ 124 (Evaluate and compare.)
Gina’s family has driven 131 miles.
Result
Gina’s family has driven 131 miles.
Page 182 Exercise 24 Answer
The perimeter is the sum of the lengths of all sides, that is
Perimeter = 3 + 5 + 3 + 3 + m.
Since the length of the perimeter is 19 cm, the expression is
19 = 3 + 5 + 3 + 3 + m.
Solve for m.
19 = 3 + 5 + 3 + 3 + m
19 = 14 + m (Find the sum.)
19 – 14 = m (Subtract 14 from both sides.)
5 = m (Find the difference.)
The length of the missing side is 5.
Result
m = 5
Page 182 Exercise 25 Answer
To find out if either of Alisa’s guesses is correct use the given equation and substitution. Substitute n for both values she gives as possible solutions.
7n = 147
7(11) = 147 (Substitute n with 11.)
77 ≠ 147
7n = 147
7(31) = 147 (Substitute n with 31.)
217 ≠ 147
Since neither 11 nor 31 are correct solutions for the given equation, both of Alisa’s guesses are wrong.
7n = 147
n = 147 ÷ 7 (Divide both sides by 7.)
n = 21
The park has 21 times as many trees as her yard.
Result
Neither of Alisa’s guesses is correct. The park has 21 times as many trees as her yard.
Page 182 Exercise 26 Answer
To find the answer use the given equation and substitution. Substitute c for the price of each item she wants to buy.
$18.62 + c = $26.00
$18.62 + $7.32 = $26.00 (Substitute c for $7.32.)
$25.94 < $26.00 (Evaluate and compare.)
$18.62 + c = $26.00
$18.62 + $7.38 = $26.00 (Substitute c for $7.38.)
$26.00 = $26.00 (Evaluate and compare.)
$18.62 + c = $26.00
$18.62 + $8.48 = $26.00 (Substitute c for $8.48.)
$27.10 > $26.00 (Evaluate and compare.)
Trish can’t buy oil paints because the total cost will then be greater than $26.00, she can buy knitting needles and she will spend less than $26.00. However, if she buys silk flowers she will spend exactly $26.00, which means the silk flowers are the most expensive item she can buy.
Result
Silk flowers.