# Solve Equations with Variables on both Sides Worksheet

Solve Equations with Variables on both Sides Worksheet
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1.
When 2 is multiplied by 2, it produces the same result as when 2 is added to 2. It is 4 in both cases. Can you think of another pair of numbers with the same arithmetical feature?
 a. 3, 3 b. 1.5, 1.5 c. 3, 4.5 d. 3, 1.5

#### Solution:

Given 2 + 2 = 2 × 2, generate a formula for this.

a + b = a × b

a = (a × b - b)
a = b(a -1)

b = aa -1

By trail and error, substitute the value of a to find b.

Substitute a = 1, b does not exist.

Substitute a = 2 then b = 2, the pair is already given.

Substitute a = 3 then b = 3 / 2 = 1.5 .

3 + 1.5 = 3 × 1.5
4.5 = 4.5
[The condition of equality satisfies.]

Therefore, another pair of numbers that satisfies the arithmetic feature is 3, 1.5.

2.
Solve: 3$x$ = 12 - $x$
 a. 2 b. 3 c. 1 d. 6

#### Solution:

3x = 12 - x
[Original equation.]

4x = 12

4x4 = 124
[Divide each side by 4.]

x = 3
[Simplify.]

The solution of the equation is x = 3.

3.
Solve:
- 3$x$ + 56 = 5$x$ + 24
 a. 3 b. 5 c. 8 d. 4

#### Solution:

- 3x + 56 = 5x + 24
[Original equation.]

- 3x + 56 + 3x = 5x + 24 + 3x

56 = 8x + 24
[Combine the like terms.]

56 - 24 = 8x + 24 - 24
[Subtract 24 from each side.]

32 = 8x
[Simplify.]

328= 88x
[Divide each side by 8.]

x = 4
[Simplify.]

4.
Find the value of $y$ in the equation 6$y$ - 2 = $y$ + 13 .
 a. 6 b. 8 c. 3

#### Solution:

6y - 2 = y + 13
[Original equation.]

6y - 2 - y = y + 13 - y
[Subtract y from each side.]

5y - 2 = 13
[Combine like terms.]

5y - 2 + 2 = 13 + 2

5y = 15
[Combine like terms.]

55y = 155
[Divide each side by 5.]

y = 3
[Simplify.]

5.
Which of the following choices is the first step in solving the equation - 2$x$ = - 32 + 19$x$?
 a. Add - 2$x$ on both sides b. Multiply 2 on both sides c. Subtract 19$x$ on both sides d. None of the above

#### Solution:

- 2x = - 32 + 19x
[Original equation.]

To have the like terms on one side, first we have to subtract 19x from each side or add 2x on each side.

6.
A Library provides two types of cards for their members: a red card, which costs $24 plus$4.25 rent per book, and a yellow card, which costs $18 plus$5.75 rent per book. For what number of books do both the cards cost the same?
 a. 4 b. 6 c. 3 d. 5

#### Solution:

Let n be the number of books for which both the cards costs same.

24 + 4.25n = 18 + 5.75n
[Write an equation.]

24 + 4.25n - 18 = 18 + 5.75n - 18
[Subtract 18 on both sides.]

6 + 4.25n = 5.75n
[Combine like terms.]

6 + 4.25n - 4.25n = 5.75n - 4.25n
[Subtract 4.25n on both sides.]

6 = 1.5n
[Simplify.]

61.5 = 1.51.5n
[Divide by 1.5 on both sides.]

4 = n
[Simplify.]

So, for 4 books both cards cost the same.

7.
Two trucks, loaded with cargo, start from Amsterdam. The first truck leaves Amsterdam traveling at a steady rate of 50 miles per hour and is 150 miles behind the second truck that travels at a steady rate of 80 miles per hour. What is the time taken by the second truck to catch up to the first truck?
 a. 5 hours b. 7 hours c. 9 hours d. None of the above

#### Solution:

Let t be the time taken by the second truck to catch up to the first truck.

Speed of the second truck x t = 150 + (speed of the first truck x t).

80t = 150 + (50t)
[Original equation.]

80t - 50t = 150 + 50t - 50t
[Subtract 50t from each side.]

30t = 150
[Combine like terms.]

30t30 = 15030
[Divide each side by 30.]

t = 5
[Simplify each side.]

The second truck will catch up to the first truck in 5 hours.

8.
Solve: 2$t$ - 4 + 45$t$ = - 2$t$ + 45
 a. 4 b. 2 c. 1 d. None of the above

#### Solution:

2t - 4 + 45t = -2t + 45
[Original equation.]

47t - 4 = -2t + 45
[Combine like terms.]

47t - 4 + 2t = -2t + 45 + 2t

49t - 4 = 45
[Simplify.]

49t - 4 + 4 = 45 + 4

49t = 49
[Simplify each side.]

49t49 = 4949
[Divide each side by 49.]

t = 1
[Simplify.]

9.
Which of the following equations has the solution of $b$ = 5?
 a. 4b+2a = 2b - 8 b. 3b - 18 - b = -b - 3 c. 17b + 5b - 2a = 7b + 8 d. -2b - 2a + 3b = 2b - 4

#### Solution:

Choices A, C and D have two variable that are a, b. Since the value of a is not considered in the data. Those choices cannot be solved.

Consider choice B, 3b - 18 - b = - b - 3.
[Original equation.]

2b - 18 = - b - 3
[Combine like terms.]

2b - 18 + b = - b - 3 + b

3b - 18 = -3
[Simplify.]

3b - 18 + 18 = -3 + 18

3b = 15
[Combine like terms.]

3b / 3 = 15 / 3
[Divide each side by 3.]

b = 5.
[Simplify.]

10.
Solve the equation 4($x$ + 2) = 4$x$ + 8 and determine whether it has one solution, no solution, or is an identity.
 a. One solution b. No solution c. Identity d. None of the above

#### Solution:

4(x + 2) = 4x + 8
[Given equation.]

4x + 8 = 4x + 8
[Multiply 4 to remove the paranthesis.]

4x + 8 - 4x = 4x + 8 - 4x
[Subtract 4x from each side.]

8 = 8
[Combine like terms.]

So, the equation is an identity.