• 52 questions

Systems of Equations Quiz

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Solve the system: x + y = 0 and 2x − y = 9.

Pick an answer, then press Check answer. Then you see Correct or Not quite, with the reason. Keys: A to D pick, Enter checks.

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

Next step Next: one solution, no solution or infinitely many

All 52 questions with answers

Solve the system 15

  • Solve the system: 2x + 5y = 1 and x + 2y = 0. Answer: (−2, 1). Source: Wikipedia, System of linear equations. Read 2026-09-23.
  • Solve the system: 2x + y = 7 and x − y = 2. Answer: (3, 1). Source: Wikipedia, System of linear equations. Read 2026-09-23.
  • Solve the system: 3x + 2y = 8 and x − y = −4. Answer: (0, 4). Source: Wikipedia, System of linear equations. Read 2026-09-23.
  • Solve the system: 3x + 4y = 10 and 3x − 2y = 4. Answer: (2, 1). Source: Wikipedia, System of linear equations. Read 2026-09-23.
  • Solve the system: 3x + y = −3 and x − y = −5. Answer: (−2, 3). Source: Wikipedia, System of linear equations. Read 2026-09-23.
  • Solve the system: 4x − y = −3 and x + y = 8. Answer: (1, 7). Source: Wikipedia, System of linear equations. Read 2026-09-23.
  • Solve the system: 5x − y = −1 and 2x + y = −6. Answer: (−1, −4). Source: Wikipedia, System of linear equations. Read 2026-09-23.
  • Solve the system: x + 2y = 1 and x − y = 10. Answer: (7, −3). Source: Wikipedia, System of linear equations. Read 2026-09-23.
  • Solve the system: x + 2y = 2 and 2x + y = 7. Answer: (4, −1). Source: Wikipedia, System of linear equations. Read 2026-09-23.
Show all 52 questions (43 more)

Solve the system, continued

  • Solve the system: x + y = 0 and 2x − y = 9. Answer: (3, −3). Source: Wikipedia, System of linear equations. Read 2026-09-23.
  • Solve the system: x + y = 7 and 3x − y = 1. Answer: (2, 5). Source: Wikipedia, System of linear equations. Read 2026-09-23.
  • Solve the system: x + y = 10 and x − y = 2. Answer: (6, 4). Source: Wikipedia, System of linear equations. Read 2026-09-23.
  • Solve the system: x + y = −5 and 2x − 3y = 0. Answer: (−3, −2). Source: Wikipedia, System of linear equations. Read 2026-09-23.
  • Solve the system: x − 2y = 1 and 2x + 3y = 16. Answer: (5, 2). Source: Wikipedia, System of linear equations. Read 2026-09-23.
  • Solve the system: x − 3y = 2 and x + y = 10. Answer: (8, 2). Source: Wikipedia, System of linear equations. Read 2026-09-23.

How many solutions? 18

  • How many solutions does the system 2x − y = 3 and x + y = 3 have? Answer: One solution. Source: Wikipedia, System of linear equations. Read 2026-09-23.
  • How many solutions does the system 3x + 2y = 6 and 6x + 4y = 12 have? Answer: Infinitely many solutions. Source: Wikipedia, System of linear equations. Read 2026-09-23.
  • How many solutions does the system 4x − 2y = 8 and 2x − y = 4 have? Answer: Infinitely many solutions. Source: Wikipedia, System of linear equations. Read 2026-09-23.
  • How many solutions does the system 5x + y = 2 and x + 5y = 2 have? Answer: One solution. Source: Wikipedia, System of linear equations. Read 2026-09-23.
  • How many solutions does the system x + y = 4 and 2x + 2y = 5 have? Answer: No solution. Source: Wikipedia, System of linear equations. Read 2026-09-23.
  • How many solutions does the system x − 3y = 2 and −2x + 6y = 1 have? Answer: No solution. Source: Wikipedia, System of linear equations. Read 2026-09-23.
  • How many solutions does the system y = 2x + 3 and y = 2x − 1 have? Answer: No solution. Source: Wikipedia, System of linear equations. Read 2026-09-23.
  • How many solutions does the system y = 2x and y = −2x + 4 have? Answer: One solution. Source: Wikipedia, System of linear equations. Read 2026-09-23.
  • How many solutions does the system y = 3x + 1 and y = −3x + 1 have? Answer: One solution. Source: Wikipedia, System of linear equations. Read 2026-09-23.
  • How many solutions does the system y = 3x + 6 and y = 3x − 6 have? Answer: No solution. Source: Wikipedia, System of linear equations. Read 2026-09-23.
  • How many solutions does the system y = 4x − 2 and y = 4x − 2 have? Answer: Infinitely many solutions. Source: Wikipedia, System of linear equations. Read 2026-09-23.
  • How many solutions does the system y = 5x + 1 and y = 5x + 1 have? Answer: Infinitely many solutions. Source: Wikipedia, System of linear equations. Read 2026-09-23.
  • How many solutions does the system y = x + 4 and y = x + 4 have? Answer: Infinitely many solutions. Source: Wikipedia, System of linear equations. Read 2026-09-23.
  • How many solutions does the system y = ½x + 2 and y = ½x − 3 have? Answer: No solution. Source: Wikipedia, System of linear equations. Read 2026-09-23.
  • How many solutions does the system y = −2x + 5 and y = 4x + 5 have? Answer: One solution. Source: Wikipedia, System of linear equations. Read 2026-09-23.
  • How many solutions does the system y = −3x + 2 and y = 2x − 3 have? Answer: One solution. Source: Wikipedia, System of linear equations. Read 2026-09-23.
  • How many solutions does the system y = −x + 7 and y = −x − 7 have? Answer: No solution. Source: Wikipedia, System of linear equations. Read 2026-09-23.
  • How many solutions does the system y = −x and y = x have? Answer: One solution. Source: Wikipedia, System of linear equations. Read 2026-09-23.

Is the point a solution? 8

  • Is (0, 2) a solution of 3x + y = 3 and x − y = 1? Answer: Neither. Source: Wikipedia, System of linear equations. Read 2026-09-23.
  • Is (1, 4) a solution of x + y = 5 and 3x − y = 0? Answer: Only the first. Source: Wikipedia, System of linear equations. Read 2026-09-23.
  • Is (2, 3) a solution of x + y = 5 and 2x − y = 1? Answer: Yes, it solves both. Source: Wikipedia, System of linear equations. Read 2026-09-23.
  • Is (2, −2) a solution of x + y = 1 and 3x + y = 5? Answer: Neither. Source: Wikipedia, System of linear equations. Read 2026-09-23.
  • Is (3, −1) a solution of 2x + y = 4 and x + 2y = 1? Answer: Only the second. Source: Wikipedia, System of linear equations. Read 2026-09-23.
  • Is (4, 1) a solution of x + y = 6 and x − y = 3? Answer: Only the second. Source: Wikipedia, System of linear equations. Read 2026-09-23.
  • Is (5, 0) a solution of x + 2y = 5 and 2x − 3y = 7? Answer: Only the first. Source: Wikipedia, System of linear equations. Read 2026-09-23.
  • Is (−1, 2) a solution of 2x + y = 0 and x + y = 1? Answer: Yes, it solves both. Source: Wikipedia, System of linear equations. Read 2026-09-23.

Word problems 6

  • A farm has chickens and cows: 12 heads and 34 legs. How many of each? Answer: 7 chickens, 5 cows. Source: Wikipedia, System of linear equations. Read 2026-09-23.
  • A quiz has 2-point and 5-point questions: 14 questions worth 40 points. How many 5-point questions? Answer: 4. Source: Wikipedia, System of linear equations. Read 2026-09-23.
  • Adult tickets cost $5 and child tickets $3. Ten tickets cost $38. How many of each were sold? Answer: 4 adult, 6 child. Source: Wikipedia, System of linear equations. Read 2026-09-23.
  • Pens cost $2 and notebooks $4. Maya buys 9 items for $26. How many notebooks? Answer: 4 notebooks. Source: Wikipedia, System of linear equations. Read 2026-09-23.
  • The sum of two numbers is 15. One is twice the other. What are they? Answer: 10 and 5. Source: Wikipedia, System of linear equations. Read 2026-09-23.
  • Two numbers add to 20 and differ by 6. What are they? Answer: 13 and 7. Source: Wikipedia, System of linear equations. Read 2026-09-23.

The ideas 5

  • A system with no solution is called… Answer: Inconsistent. Source: Wikipedia, System of linear equations. Read 2026-09-23.
  • In the system y = 2x + 1 and 3x + y = 11, which method is quickest to start? Answer: Substitute 2x + 1 for y. Source: Wikipedia, System of linear equations. Read 2026-09-23.
  • On a graph, what is the solution of a system of two linear equations? Answer: The point where the two lines cross. Source: Wikipedia, System of linear equations. Read 2026-09-23.
  • The two equations of a system describe the same line. How many solutions? Answer: Infinitely many solutions. Source: Wikipedia, System of linear equations. Read 2026-09-23.
  • Two lines have the same slope and different y-intercepts. How many solutions does the system have? Answer: No solution. Source: Wikipedia, System of linear equations. Read 2026-09-23.

How this list was compiled: Questions, wrong choices and explanations were written for this site. The three cases (one solution, no solution, infinitely many), the graph reading (the solution is where the lines meet) and the words consistent and inconsistent were checked on Wikipedia's System of linear equations (https://en.wikipedia.org/wiki/System_of_linear_equations), read 2026-09-23. Every system, answer and wrong choice was checked by solving the system exactly. Licence: own work

How it works

This systems of equations quiz has 52 questions on pairs of linear equations in x and y: solve the system, say how many solutions it has, test a point, and set up word problems.

Read the full guide

Five sets

Solve the system gives four ordered pairs to choose from. How many solutions? compares slopes and intercepts. Is the point a solution? substitutes into both equations. Word problems cover tickets, coins and legs. The ideas set checks the vocabulary.

Before a test

Practice brings a missed question back until you get it right. After a round, Print lists its questions: remove one, add more up to all 52, and tick With answer key.

  • Check an answer by putting it back into both equations; a pair that fits only one is not the solution.
  • Same slope and different intercepts: parallel lines, no solution. Same slope and same intercept: the same line.

Questions and answers

How do you solve a system of equations?

Substitution: solve one equation for a variable and put it into the other. Elimination: add or subtract the equations so one variable cancels. Graphing: the solution is where the two lines cross.

How can you tell how many solutions a system has?

Write both lines as y = mx + b. Different slopes cross once. The same slope with different intercepts gives parallel lines and no solution. The same slope and intercept is one line with infinitely many solutions.

What is an inconsistent system?

A system with no solution, such as x + y = 4 and 2x + 2y = 5. A system with at least one solution is consistent.

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