One Solution No Solution Infinite Solutions Worksheet

Print the worksheet, or press Quiz me on screen and tap how many solutions the equation has.

See all 50 with answers
  1. 1.How many solutions does 3x + 5 = 3x + 5 have?
  2. 2.How many solutions does 2x + 7 = 2x − 4 have?
  3. 3.How many solutions does 5x − 1 = 3x + 9 have?
  4. 4.How many solutions does x + 8 = x + 8 have?
  5. 5.How many solutions does 4x = 4x + 1 have?
  6. 6.How many solutions does 6x − 2 = 6x − 2 have?
  7. 7.How many solutions does 7x + 3 = 2x − 12 have?
  8. 8.How many solutions does −2x + 9 = −2x + 9 have?
  9. 9.How many solutions does 9 − x = −x + 4 have?
  10. 10.How many solutions does x − 6 = 2x + 1 have?
  11. 11.How many solutions does 10x = 10x have?
  12. 12.How many solutions does 5 + 4x = 4x + 5 have?
  13. 13.How many solutions does 3x − 8 = 3x + 8 have?
  14. 14.How many solutions does x = −x have?
  15. 15.How many solutions does 12 − 3x = 3x have?
  16. 16.How many solutions does 2(x + 3) = 2x + 6 have?
  17. 17.How many solutions does 3(x − 4) = 3x − 4 have?
  18. 18.How many solutions does 4(2x + 1) = 8x + 4 have?
  19. 19.How many solutions does 5(x + 2) = 3x + 20 have?
  20. 20.How many solutions does 2(3x − 1) = 6x + 2 have?
  21. 21.How many solutions does −3(x − 2) = −3x + 6 have?
  22. 22.How many solutions does 6x + 4 = 2(3x + 2) have?
  23. 23.How many solutions does 4(x − 1) + 2 = 4x − 2 have?
  24. 24.How many solutions does 3(x + 1) + x = 4x + 5 have?
  25. 25.How many solutions does 2(x + 5) = 5(x − 1) have?
  26. 26.How many solutions does 7x − 2(x + 1) = 5x − 2 have?
  27. 27.How many solutions does 3(2x + 4) = 2(3x + 6) have?
  28. 28.How many solutions does 5(x − 3) = 5x − 3 have?
  29. 29.How many solutions does x + 2(x − 3) = 3x + 6 have?
  30. 30.How many solutions does 8 − 2(x − 4) = −2x + 16 have?
  31. 31.How many solutions does 4(x + 3) − x = 3(x + 5) have?
  32. 32.How many solutions does 3(x − 2) = x + 4 have?
  33. 33.How many solutions does 2(4 − x) = 3(x + 1) have?
  34. 34.Which value of k gives 4x + 7 = 4x + k infinitely many solutions?
  35. 35.Which value of k gives kx + 7 = 4x + 1 no solution?
  36. 36.Which value of k gives kx + 3 = 5x − 2 no solution?
  37. 37.Which value of k gives kx + 3 = 5x + 3 infinitely many solutions?
  38. 38.Which value of k gives 2(x + k) = 2x + 10 infinitely many solutions?
  39. 39.Which value of k gives 3(x − k) = 3x − 12 infinitely many solutions?
  40. 40.Which value of k gives kx − 6 = 2x + 1 no solution?
  41. 41.Which value of k gives 6x + 1 = k(2x + 1) no solution?
  42. 42.Which value of k gives k(x + 2) = 4x + 8 infinitely many solutions?
  43. 43.Which value of k gives 5x − k = 5x + 3 infinitely many solutions?
  44. 44.Which value of k gives 2(kx + 1) = 8x + 2 infinitely many solutions?
  45. 45.You simplify an equation and the x terms cancel, leaving 5 = 5. How many solutions are there?
  46. 46.You simplify an equation and the x terms cancel, leaving 3 = −1. How many solutions are there?
  47. 47.After simplifying, the two sides have different x terms, such as 4x and 2x. How many solutions are there?
  48. 48.After simplifying, both sides have the same x term but different numbers. How many solutions are there?
  49. 49.Which is another name for an equation that is true for every value of x?
  50. 50.A linear equation ax + b = 0 has exactly one solution when…
How many solutions does 3x − 8 = 3x + 8 have?

Print covers the next 20 questions. Share this set copies a link to the same questions in the same order.

Exam mode: 20 questions, no time limit.

More options

Unofficial practice questions, not affiliated with any school, textbook publisher or worksheet maker. Bank checked September 23, 2026.

Next step Another maths drill: fill in the unit circleNext: more algebra practice with trig identities

How it works

This worksheet has 50 linear equations to sort by how many solutions each has: one, none or infinitely many. Every answer shows both sides simplified, so you can see why.

Read the full guide

Four sets

Simplified equations are ready to compare. Distribute first has brackets to multiply out. Find the value of k asks which number makes an equation have no solution or infinitely many. The rules set checks the shortcut itself.

Print or practise

Practice brings missed questions back until you get them right. The page opens on its worksheet of all 50: remove any, tick With answer key, then press Print the worksheet.

  • Simplify each side, then compare the x terms. Different x terms: one solution. Same x term: look at the numbers.
  • Same x term and same number means infinitely many solutions; same x term and different numbers means no solution.

Questions and answers

How do you know if an equation has one solution, no solution or infinitely many?

Simplify both sides to the form ax + b. If the x coefficients differ, there is one solution. If they match and the constants match, every x works. If they match and the constants differ, no x works.

What does 5 = 5 mean when solving an equation?

The x terms cancelled and left a true statement, so every number is a solution: infinitely many. If they leave a false statement such as 3 = −1, there is no solution.

Is there an answer key?

Yes. Tick With answer key and the printed worksheet ends with its key; the list under the tool, All 50 questions with answers, shows the working.

All 50 questions with answers

Simplified equations 15

  • How many solutions does 2x + 7 = 2x − 4 have? Answer: No solution. Source: Wikipedia, Linear equation. Read 2026-09-23.
  • How many solutions does 3x + 5 = 3x + 5 have? Answer: Infinitely many solutions. Source: Wikipedia, Linear equation. Read 2026-09-23.
  • How many solutions does 3x − 8 = 3x + 8 have? Answer: No solution. Source: Wikipedia, Linear equation. Read 2026-09-23.
  • How many solutions does 4x = 4x + 1 have? Answer: No solution. Source: Wikipedia, Linear equation. Read 2026-09-23.
  • How many solutions does 5 + 4x = 4x + 5 have? Answer: Infinitely many solutions. Source: Wikipedia, Linear equation. Read 2026-09-23.
  • How many solutions does 5x − 1 = 3x + 9 have? Answer: One solution. Source: Wikipedia, Linear equation. Read 2026-09-23.
  • How many solutions does 6x − 2 = 6x − 2 have? Answer: Infinitely many solutions. Source: Wikipedia, Linear equation. Read 2026-09-23.
  • How many solutions does 7x + 3 = 2x − 12 have? Answer: One solution. Source: Wikipedia, Linear equation. Read 2026-09-23.
  • How many solutions does 9 − x = −x + 4 have? Answer: No solution. Source: Wikipedia, Linear equation. Read 2026-09-23.
  • How many solutions does 10x = 10x have? Answer: Infinitely many solutions. Source: Wikipedia, Linear equation. Read 2026-09-23.
  • How many solutions does 12 − 3x = 3x have? Answer: One solution. Source: Wikipedia, Linear equation. Read 2026-09-23.
  • How many solutions does x + 8 = x + 8 have? Answer: Infinitely many solutions. Source: Wikipedia, Linear equation. Read 2026-09-23.
  • How many solutions does x = −x have? Answer: One solution. Source: Wikipedia, Linear equation. Read 2026-09-23.
  • How many solutions does x − 6 = 2x + 1 have? Answer: One solution. Source: Wikipedia, Linear equation. Read 2026-09-23.
  • How many solutions does −2x + 9 = −2x + 9 have? Answer: Infinitely many solutions. Source: Wikipedia, Linear equation. Read 2026-09-23.
Show all 50 questions (35 more)

Distribute first 18

  • How many solutions does 2(3x − 1) = 6x + 2 have? Answer: No solution. Source: Wikipedia, Linear equation. Read 2026-09-23.
  • How many solutions does 2(4 − x) = 3(x + 1) have? Answer: One solution. Source: Wikipedia, Linear equation. Read 2026-09-23.
  • How many solutions does 2(x + 3) = 2x + 6 have? Answer: Infinitely many solutions. Source: Wikipedia, Linear equation. Read 2026-09-23.
  • How many solutions does 2(x + 5) = 5(x − 1) have? Answer: One solution. Source: Wikipedia, Linear equation. Read 2026-09-23.
  • How many solutions does 3(2x + 4) = 2(3x + 6) have? Answer: Infinitely many solutions. Source: Wikipedia, Linear equation. Read 2026-09-23.
  • How many solutions does 3(x + 1) + x = 4x + 5 have? Answer: No solution. Source: Wikipedia, Linear equation. Read 2026-09-23.
  • How many solutions does 3(x − 2) = x + 4 have? Answer: One solution. Source: Wikipedia, Linear equation. Read 2026-09-23.
  • How many solutions does 3(x − 4) = 3x − 4 have? Answer: No solution. Source: Wikipedia, Linear equation. Read 2026-09-23.
  • How many solutions does 4(2x + 1) = 8x + 4 have? Answer: Infinitely many solutions. Source: Wikipedia, Linear equation. Read 2026-09-23.
  • How many solutions does 4(x + 3) − x = 3(x + 5) have? Answer: No solution. Source: Wikipedia, Linear equation. Read 2026-09-23.
  • How many solutions does 4(x − 1) + 2 = 4x − 2 have? Answer: Infinitely many solutions. Source: Wikipedia, Linear equation. Read 2026-09-23.
  • How many solutions does 5(x + 2) = 3x + 20 have? Answer: One solution. Source: Wikipedia, Linear equation. Read 2026-09-23.
  • How many solutions does 5(x − 3) = 5x − 3 have? Answer: No solution. Source: Wikipedia, Linear equation. Read 2026-09-23.
  • How many solutions does 6x + 4 = 2(3x + 2) have? Answer: Infinitely many solutions. Source: Wikipedia, Linear equation. Read 2026-09-23.
  • How many solutions does 7x − 2(x + 1) = 5x − 2 have? Answer: Infinitely many solutions. Source: Wikipedia, Linear equation. Read 2026-09-23.
  • How many solutions does 8 − 2(x − 4) = −2x + 16 have? Answer: Infinitely many solutions. Source: Wikipedia, Linear equation. Read 2026-09-23.
  • How many solutions does x + 2(x − 3) = 3x + 6 have? Answer: No solution. Source: Wikipedia, Linear equation. Read 2026-09-23.
  • How many solutions does −3(x − 2) = −3x + 6 have? Answer: Infinitely many solutions. Source: Wikipedia, Linear equation. Read 2026-09-23.

Find the value of k 11

  • Which value of k gives 2(kx + 1) = 8x + 2 infinitely many solutions? Answer: 4. Source: Wikipedia, Linear equation. Read 2026-09-23.
  • Which value of k gives 2(x + k) = 2x + 10 infinitely many solutions? Answer: 5. Source: Wikipedia, Linear equation. Read 2026-09-23.
  • Which value of k gives 3(x − k) = 3x − 12 infinitely many solutions? Answer: 4. Source: Wikipedia, Linear equation. Read 2026-09-23.
  • Which value of k gives 4x + 7 = 4x + k infinitely many solutions? Answer: 7. Source: Wikipedia, Linear equation. Read 2026-09-23.
  • Which value of k gives 5x − k = 5x + 3 infinitely many solutions? Answer: −3. Source: Wikipedia, Linear equation. Read 2026-09-23.
  • Which value of k gives 6x + 1 = k(2x + 1) no solution? Answer: 3. Source: Wikipedia, Linear equation. Read 2026-09-23.
  • Which value of k gives k(x + 2) = 4x + 8 infinitely many solutions? Answer: 4. Source: Wikipedia, Linear equation. Read 2026-09-23.
  • Which value of k gives kx + 3 = 5x + 3 infinitely many solutions? Answer: 5. Source: Wikipedia, Linear equation. Read 2026-09-23.
  • Which value of k gives kx + 3 = 5x − 2 no solution? Answer: 5. Source: Wikipedia, Linear equation. Read 2026-09-23.
  • Which value of k gives kx + 7 = 4x + 1 no solution? Answer: 4. Source: Wikipedia, Linear equation. Read 2026-09-23.
  • Which value of k gives kx − 6 = 2x + 1 no solution? Answer: 2. Source: Wikipedia, Linear equation. Read 2026-09-23.

The rules 6

  • A linear equation ax + b = 0 has exactly one solution when… Answer: a is not 0. Source: Wikipedia, Linear equation. Read 2026-09-23.
  • After simplifying, both sides have the same x term but different numbers. How many solutions are there? Answer: No solution. Source: Wikipedia, Linear equation. Read 2026-09-23.
  • After simplifying, the two sides have different x terms, such as 4x and 2x. How many solutions are there? Answer: One solution. Source: Wikipedia, Linear equation. Read 2026-09-23.
  • Which is another name for an equation that is true for every value of x? Answer: An identity. Source: Wikipedia, Identity (mathematics). Read 2026-09-23.
  • You simplify an equation and the x terms cancel, leaving 3 = −1. How many solutions are there? Answer: No solution. Source: Wikipedia, Linear equation. Read 2026-09-23.
  • You simplify an equation and the x terms cancel, leaving 5 = 5. How many solutions are there? Answer: Infinitely many solutions. Source: Wikipedia, Linear equation. Read 2026-09-23.

How this list was compiled: Questions, wrong choices and explanations were written for this site. The rule (ax + b = 0 has one solution when a ≠ 0; when a = 0 it has no solution if b ≠ 0 and every number is a solution if b = 0) was checked on Wikipedia's Linear equation article (https://en.wikipedia.org/wiki/Linear_equation), read 2026-09-23. Every equation, answer and value of k was checked by simplifying both sides. Licence: own work

More quizzes

Show 6 more

Browse Science Quizzes