➕ Algebra 1 · Equations

Equations with Variables on Both Sides practice questions

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Answer key with explanations

1. Solve 8x + 3 = 5x + 21.

   Answer: x = 6. Subtract 5x from both sides: 3x + 3 = 21. Subtract 3: 3x = 18. Divide by 3: x = 6. Check: 8(6) + 3 = 51 and 5(6) + 21 = 51.

2. Solve 2x − 9 = 5x + 6.

   Answer: x = −5. Subtract 2x from both sides to keep the variable on the right: −9 = 3x + 6. Subtract 6: −15 = 3x. Divide by 3: x = −5. Check: 2(−5) − 9 = −19 and 5(−5) + 6 = −19.

3. Solve 3(x + 2) = 2x + 11.

   Answer: x = 5. Distribute: 3x + 6 = 2x + 11. Subtract 2x: x + 6 = 11. Subtract 6: x = 5. Check: 3(5 + 2) = 21 and 2(5) + 11 = 21.

4. How many solutions does 4x + 7 = 4(x + 2) have?

   Answer: No solution. Distribute: 4(x + 2) = 4x + 8, so the equation is 4x + 7 = 4x + 8. Subtract 4x from both sides: 7 = 8, which is false. No value of x works, so there is no solution.

5. Which statement describes the solutions of 3(2x − 1) = 6x − 3?

   Answer: Every number is a solution.. Distribute: 3(2x − 1) = 6x − 3, which is exactly the right side. Subtract 6x from both sides: −3 = −3, which is true for every x. So every number is a solution.

6. For which value of k does 3x + 8 = 3x + k have infinitely many solutions?

   Answer: k = 8. Subtract 3x from both sides: 8 = k. If k = 8, the leftover 8 = 8 is true for every x. For any other value of k the leftover is false, so there is no solution.

7. Solve x/4 − 3 = x/2 + 1.

   Answer: x = −16. Multiply every term on both sides by 4: x − 12 = 2x + 4. Subtract x: −12 = x + 4. Subtract 4: x = −16. Check: −16/4 − 3 = −7 and −16/2 + 1 = −7.

8. Rink A charges 8 dollars to get in plus 3 dollars per hour for skates. Rink B charges 2 dollars to get in plus 5 dollars per hour for skates. After how many hours of skating is the total cost the same at both rinks?

   Answer: 3 hours. After h hours, Rink A costs 8 + 3h dollars and Rink B costs 2 + 5h dollars. Set them equal: 8 + 3h = 2 + 5h. Subtract 3h: 8 = 2 + 2h. Subtract 2: 6 = 2h. Divide by 2: h = 3. Check: Rink A costs 8 + 9 = 17 and Rink B costs 2 + 15 = 17 dollars.

9. Solve 5 − 3(x − 1) = 2x − 12.

   Answer: x = 4. Distribute −3: 5 − 3x + 3 = 2x − 12, so 8 − 3x = 2x − 12. Add 3x to both sides: 8 = 5x − 12. Add 12: 20 = 5x. Divide by 5: x = 4. Check: 5 − 3(3) = −4 and 2(4) − 12 = −4.