Solving Equations with Variables on both Sides Worksheet

Solving Equations with Variables on both Sides Worksheet
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1.
Paul wants to go to Atlanta from San Francisco. He is comparing the rental plans of two companies. Henry Car Rentals charges $110 per week plus 60 cents per mile. William Car Rentals charges$90 per week plus 64 cents per mile. Find the number of miles for which the rental charges of the two companies are the same.
 a. 825 miles b. 600 miles c. 500 miles d. 679 miles

Solution:

Let the number of miles be x so that the rental charges are same.

Weekly charges for hiring car from Henry Car Rentals for x miles = $(110 + 0.6x) Weekly charges for hiring car from William Car Rentals for x miles =$ (90 + 0.64x)

110 + 0.6 x = 90 + 0.64x
[Equate the rental charges.]

20 = 0.04x

500 = x
[Solve for x.]

So, the car rental charges for the two companies are same for 500 miles.

2.
Solve:
20$y$ - 7$y$ + 15 = 8$y$ + 30
 a. 3 b. 2 c. 5 d. 4

Solution:

20y - 7y + 15 = 8y + 30
[Original equation.]

13y + 15 = 8y + 30
[Combine like terms.]

13y + 15 - 15 = 8y + 30 - 15
[Subtract 15 from each side.]

13y = 8y + 15
[Simplify both sides.]

13y - 8y = 8y + 15 - 8y
[Subtract 8y from each side.]

5y = 15
[Simplify both sides.]

y = 3
[Simplify.]

3.
Solve: $\frac{j}{9}$ + 14.93 = 9($j$ - 2)
 a. 6.7 b. 7.7 c. 3.7 d. 5.7

Solution:

j9 + 14.93 = 9(j - 2)
[Original equation.]

j9 + 14.93 = 9(j) - 9(2)
[Use distributive property.]

j9 + 14.93 + 18 = 9j - 18 + 18

j9 + 32.93 = 9j
[Combine like terms.]

(j9 + 32.93) × 9 = 9j × 9
[Multiply each side by 9.]

j + 296.37 = 81j
[Simplify.]

j + 296.37 - j = 81j - j
[Subtract j from each side.]

296.37 = 80j
[Simplify.]

296.37 / 80 = 80j80
[Divide each side by 80.]

j = 3.7
[Simplify.]

4.
Solve 4$x$ = 50 - $x$.
 a. 13 b. 8 c. 10 d. 9

Solution:

4x = 50 - x
[Original equation.]

5x = 50

5x5 = 505
[Divide each side by 5.]

x = 10
[Simplify.]

The solution of the equation is x = 10.

5.
Solve:
- 2$x$ + 55 = 8$x$ + 35
 a. 4 b. 1 c. 3 d. 2

Solution:

- 2x + 55 = 8x + 35
[Original equation.]

- 2x + 55 + 2x = 8x + 35 + 2x

55 = 10x + 35
[Combine the like terms.]

55 - 35 = 10x + 35 - 35
[Subtract 35 from each side.]

20 = 10x
[Simplify.]

2010 = 10x10
[Divide each side by 10.]

x = 2
[Simplify.]

6.
Solve the equation.
4$y$ - 3 = $y$ + 3
 a. 7 b. 1 c. 3 d. 2

Solution:

4y - 3 = y + 3
[Original equation.]

4y - 3 - y = y + 3 - y
[Subtract y from each side.]

3y - 3 = 3
[Combine like terms.]

3y - 3 + 3 = 3 + 3

3y = 6
[Combine like terms.]

3y3 = 63
[Divide each side by 3.]

y = 2
[Simplify.]

7.
Choose the first step that would help you solve the equation - 7$x$ = - 35 + 20$x$.
 a. add - 7$x$ to both sides b. subtract 20$x$ from both sides c. multiply both sides by 7 d. divide both sides by 7

Solution:

- 7x = - 35 + 20x
[Original equation.]

To have the like terms on one side, first we need to subtract 20x from each side or add 7x to each side.

8.
A library provides two types of cards for its members. A red card that costs $48 plus$4.25 rent per book and a yellow card costs $36 plus$5.75 rent per book. Find the number of books for which both cards would cost the same.
 a. 9.3 b. 8 c. 1 d. 10.5

Solution:

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

48 + 4.25n = 36 + 5.75n
[Write an equation.]

48 + 4.25n - 36 = 36 + 5.75n - 36
[Subtract 36 on both sides.]

12 + 4.25n = 5.75n
[Combine like terms.]

12 + 4.25n - 4.25n = 5.75n - 4.25n
[Subtract 4.25n on both sides.]

12 = 1.5n
[Simplify.]

121.5 = 1.5n1.5
[Divide by 1.5 on both sides.]

8 = n
[Simplify.]

So, for 8 books both cards cost the same.

9.
Solve the equation, 2($x$ + 3) = 2$x$ + 6 and determine whether it has one solution, no solution, or is an identity.
 a. an identity b. not enough information given c. one solution d. no solution

Solution:

2(x + 3) = 2x + 6
[Given equation.]

2x + 6 = 2x + 6
[Multiply 2 to remove the paranthesis.]

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

6 = 6
[Combine like terms.]

So the equation is an identity.

10.
Solve the equation 2($x$ + 9) = 2$x$ + 6 and determine whether it has one solution, no solution or is an identity.
 a. not enough information given b. one solution c. no solution d. an identity

Solution:

2(x + 9) = 2x + 6
[Original equation.]

2x + 18 = 2x + 6
[Distribute 2.]

2x + 18 - 2x = 2x + 6 - 2x
[Subtract 2x from each side.]

18 ¹ 6
[Combine like terms.]

The equation 18 = 6 is never true. So, the original equation has no solution.