SOLVING EQUATIONS WITH VARIABLES ON BOTH SIDES WORKSHEET WITH ANSWER KEY

Problem 1 :

Solve for r : 

6r + 7  =  13 + 7r

Problem 2 :

Solve for x : 

-7x - 3x + 2  =  -8x - 8

Problem 3 :

Solve for n : 

4n - 40  =  7(-2n + 2) 

Problem 4 :

Solve for k :

5(k - 3) - 7(6 - k)  =  24 - 3(8 - k) - 3

Problem 5 :

Solve for x : 

3(1 - 3x)  =  2(-4x + 7)

Problem 6 :

Solve for x :  

⁴ˣ⁄₅ - ⁷⁄₄  =  ˣ⁄₅ + ˣ⁄₄

Problem 7 :

Solve for x :  

⁽ˣ ⁻ ²⁾⁄₂ + ⁽ˣ ⁺ ¹⁰⁾⁄₉  =  5

Problem 8 : 

Solve for x : 

(x + 1)(2x + 1)  =  (x + 3)(2x + 3) - 14

Problem 9 : 

Solve the following equation : 

½(8y - 6) = 5y - (y + 3)

Problem 10 : 

Solve the following equation : 

2(1 - x) + 5x  =  3(x + 1)

Answers

1. Answer :

6r + 7  =  13 + 7r

Subtract 6r from each side.

7  =  13 + r

Subtract 13 from each side.

-6  =  r

2. Answer :

-7x - 3x + 2  =  -8x - 8

Simplify.

-10x + 2  =  -8x - 8

Add 10x to each side. 

2  =  2x - 8

Add 8 to each side.

10  =  2x

Divide each side by 2.

5  =  x

3. Answer :

4n - 40  =  7(-2n + 2)

Use distributive property. 

4n - 40  =  -14n + 14

Add 14n to each side. 

18n - 40  =  14

Add 40 to each side. 

18n  =  54

Divide each side by 18.

n  =  3

4. Answer :

5(k - 3) - 7(6 - k)  =  24 - 3(8 - k) - 3

Use distributive property. 

5k - 15 - 42 + 7k  =  24 - 24 + 3k - 3

Simplify. 

12k - 57  =  3k - 3

Subtract 3k from each side. 

9k - 57  =  -3

Add 57 to each side.

9k  =  54

Divide each side by 9.

k  =  6

5. Answer :

3(1 - 3x)  =  2(-4x + 7)

Use distributive property. 

3 - 9x  =  -8x + 14

Add 9x to each side.

3  =  x + 14

Subtract 14 from each side.

-11  =  x

6. Answer :

⁴ˣ⁄₅ - ⁷⁄₄  =  ˣ⁄₅ + ˣ⁄₄

The least common multiple of the denominators in the equation is 4 × 5  =  20 and we proceed as follows :

20(⁴ˣ⁄₅ - ⁷⁄₄) = 20(ˣ⁄₅ + ˣ⁄₄)

20(⁴ˣ⁄₅) - 20(⁷⁄₄) = 20(ˣ⁄₅) + 20(ˣ⁄₄)

16x - 35 = 4x + 5x

16x - 35 = 9x

Subtract 9x from each side. 

7x - 35 = 0

Add 35 to each side.

7x = 35

Divide each side by 7.

x = 5

7. Answer :

⁽ˣ ⁻ ²⁾⁄₂ + ⁽ˣ ⁺ ¹⁰⁾⁄₉ = 5

The least common multiple of the denominators in the equation is 2 × 9  =  18 and we proceed as follows :

18(⁽ˣ ⁻ ²⁾⁄₂ + ⁽ˣ ⁺ ¹⁰⁾⁄₉) = 18(5)

18(⁽ˣ ⁻ ²⁾⁄₂) + 18(⁽ˣ ⁺ ¹⁰⁾⁄₉) = 90

9(x - 2) + 2(x + 10) = 90

9x - 18 + 2x + 20 = 90

11x + 2 = 90

Subtract 2 from each side. 

11x = 88

Divide each side by 11. 

x = 8

8. Answer :

(x + 1)(2x + 1)  =  (x + 3)(2x + 3) - 14

Simplify. 

2x² + 3x + 1  =  2x² + 9x + 9 - 14

2x² + 3x + 1  =  2x² + 9x - 5

Subtract 2x² from each side. 

3x + 1  =  9x - 5

Subtract 3x from each side. 

1  =  6x - 5

Add 5 to each side.

6  =  6x

Divide each side by 6.

1  =  x

9. Answer :

½(8y - 6) = 5y - (y + 3)

Simplify both sides. 

½(8y) - ½(6) = 5y - (y + 3)

4y - 3 = 5y - y - 3

4y - 3 = 4y - 3  

Subtract 4y from each side. 

-3 = -3

The above result is true. Because the result we get at the last step is true, the given equation has infinitely has many solutions. 

10. Answer :

2(1 - x) + 5x  =  3(x + 1)

Simplify both sides. 

2 - 2x + 5x  =  3x + 3

2 + 3x  =  3x + 3

Subtract 3x from each side. 

2  =  3

The above result is false. Because 2 is not equal to 3. Because the result we get at the last step is false, the given equation has no solution.  

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