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 :
(1/2)(8y - 6) = 5y - (y + 3)
Problem 10 :
Solve the following equation :
2(1 - x) + 5x = 3(x + 1)
Detailed Answer Key
Problem 1 :
Solve for r :
6r + 7 = 13 + 7r
Solution :
6r + 7 = 13 + 7r
Subtract 6r from each side.
7 = 13 + r
Subtract 13 from each side.
-6 = r
Problem 2 :
Solve for x :
-7x - 3x + 2 = -8x - 8
Solution :
-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
Problem 3 :
Solve for n :
4n - 40 = 7(-2n + 2)
Solution :
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
Problem 4 :
Solve for k :
5(k - 3) - 7(6 - k) = 24 - 3(8 - k) - 3
Solution :
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
Problem 5 :
Solve for x :
3(1 - 3x) = 2(-4x + 7)
Solution :
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
Problem 6 :
Solve for x :
(4x/5) - (7/4) = (x/5) + (x/4)
Solution :
(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
Problem 7 :
Solve for x :
(x - 2)/2 + (x + 10)/9 = 5
Solution :
(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 = 4
Problem 8 :
Solve for x :
(x + 1)(2x + 1) = (x + 3)(2x + 3) - 14
Solution :
(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
Problem 9 :
Solve the following equation :
(1/2)(8y - 6) = 5y - (y + 3)
Solution :
(1/2)(8y - 6) = 5y - (y + 3)
Simplify both sides.
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.
Problem 10 :
Solve the following equation :
2(1 - x) + 5x = 3(x + 1)
Solution :
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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