Special Solutions - Worksheet 9 - 8th Grade

Hard

Distinguish equations with no solution from identities true for every real number.

hard: Use another set to practice explaining and checking your answers.

23 problems · 15-20 minutes · Skills: algebra, critical thinking, equation solving

About This Worksheet

This hard-level worksheet contains 23 problems designed for 8th Grade students practicing special solutions: no solution & infinite solutions. Hard worksheets feature challenging problems including multi-step questions, missing values, and real-world applications to push students further.

Use signed arithmetic, substitute values and read expressions with variables.

  • Distinguish equations with no solution from identities true for every real number.
  • Check your result using the method in the example before moving to the next question.

How to Solve: Worked Example

Example Problem: Solve 6(x + 7) = 6x + 42. State whether there are no solutions or infinitely many solutions.
Solution Steps:
  1. Expand the left side to 6x + 42.
  2. Subtract the x-terms from both sides.
  3. The remaining equality is true for every x.
Answer: Infinitely many solutions (all real x).
Tip: Check your result using the method in the example before moving to the next question.
1
Solve 4(x + 1) = 4x + 4. State whether there are no solutions or infinitely many solutions.
2
Solve 12(x + 1) = 12x + 13. State whether there are no solutions or infinitely many solutions.
3
Solve 6(x + 4) = 6x + 24. State whether there are no solutions or infinitely many solutions.
4
Solve 4(x + 8) = 4x + 33. State whether there are no solutions or infinitely many solutions.
5
Solve 2(x + 1) = 2x + 2. State whether there are no solutions or infinitely many solutions.
6
Solve 11(x + 9) = 11x + 100. State whether there are no solutions or infinitely many solutions.
7
Solve 6(x + 5) = 6x + 30. State whether there are no solutions or infinitely many solutions.
8
Solve 2(x + 7) = 2x + 15. State whether there are no solutions or infinitely many solutions.
9
Solve 2(x + 5) = 2x + 10. State whether there are no solutions or infinitely many solutions.
10
Solve 2(x + 6) = 2x + 13. State whether there are no solutions or infinitely many solutions.
11
Solve 10(x + 6) = 10x + 60. State whether there are no solutions or infinitely many solutions.
12
Solve 6(x + 9) = 6x + 55. State whether there are no solutions or infinitely many solutions.
13
Solve 2(x + 3) = 2x + 6. State whether there are no solutions or infinitely many solutions.
14
Solve 6(x + 5) = 6x + 31. State whether there are no solutions or infinitely many solutions.
15
Solve 10(x + 7) = 10x + 70. State whether there are no solutions or infinitely many solutions.
16
Solve 8(x + 4) = 8x + 33. State whether there are no solutions or infinitely many solutions.
17
Solve 2(x + 8) = 2x + 16. State whether there are no solutions or infinitely many solutions.
18
Solve 8(x + 9) = 8x + 73. State whether there are no solutions or infinitely many solutions.
19
Solve 5(x + 6) = 5x + 30. State whether there are no solutions or infinitely many solutions.
20
Solve 2(x + 8) = 2x + 17. State whether there are no solutions or infinitely many solutions.
21
Solve 10(x + 1) = 10x + 10. State whether there are no solutions or infinitely many solutions.
22
Solve 8(x + 8) = 8x + 65. State whether there are no solutions or infinitely many solutions.
23
Solve 7(x + 2) = 7x + 14. State whether there are no solutions or infinitely many solutions.

Answer Key

1. Infinitely many solutions (all real x).2. No solution.3. Infinitely many solutions (all real x).4. No solution.5. Infinitely many solutions (all real x).6. No solution.7. Infinitely many solutions (all real x).8. No solution.9. Infinitely many solutions (all real x).10. No solution.11. Infinitely many solutions (all real x).12. No solution.13. Infinitely many solutions (all real x).14. No solution.15. Infinitely many solutions (all real x).16. No solution.17. Infinitely many solutions (all real x).18. No solution.19. Infinitely many solutions (all real x).20. No solution.21. Infinitely many solutions (all real x).22. No solution.23. Infinitely many solutions (all real x).

Frequently Asked Questions

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How do I find worksheets at different difficulty levels?

Each subtopic has 10 worksheets: Worksheets 1-3 are Easy, 4-7 are Medium, and 8-10 are Hard. Start with Easy worksheets to build confidence, then progress to harder ones.

What skills does this linear equations worksheet cover?

This worksheet focuses on algebra, critical thinking, equation solving. It is designed for 8th Grade students and provides structured practice through varied problem types with a worked example to guide students before they begin.

Can I use this worksheet in my classroom?

Yes. All worksheets on K12Worksheets are free for classroom use. You can print copies for your entire class. With 10 worksheets per subtopic at three difficulty levels, you have plenty of options for differentiated instruction and individualized practice.

How many problems are on this worksheet?

This worksheet contains 23 problems. It typically takes 15-20 minutes to complete. An answer key with all solutions is included so parents and teachers can check work quickly.