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Math Practice

Systems of Equations Practice Problems

Start by testing whether a point solves a system, then decide how many solutions a system has without solving it, and finish by finding the solution of systems where both equations are already solved for y.

  1. 1.

    Is a solution of and ? Answer "yes" or "no".

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    ✓ and ✓. It satisfies both equations, so it is a solution.

  2. 2.

    Is a solution of and ? Answer "yes" or "no".

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    ✓, but . It fails the second equation, so it is not a solution.

  3. 3.

    How many solutions does the system and have? Answer 0, 1 or "infinitely many".

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    Same slope (3), different intercepts (−2 and 5): the lines are parallel, so there is no solution.

  4. 4.

    How many solutions does the system and have? Answer 0, 1 or "infinitely many".

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    The slopes 2 and −1 are different, so the lines cross exactly once.

  5. 5.

    How many solutions does and have? Answer 0, 1 or "infinitely many".

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    Every coefficient ratio is . The first equation is the second doubled, so they are the same line.

  6. 6.

    How many solutions does and have? Answer 0, 1 or "infinitely many".

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    but . Same slope, different intercepts: parallel lines, no solution.

  7. 7.

    Solve and .

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    Set the expressions equal: , so and . Then .

  8. 8.

    Solve and .

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    gives , so and .

  9. 9.

    For which value of does the system and have no solution?

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    No solution means parallel lines: equal slopes with different intercepts. The intercepts 1 and −2 already differ, so .

  10. 10.

    For which value of does and have infinitely many solutions?

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    The second equation must be the first one doubled: , so .

  11. 11.

    If is the solution of and , what is ?

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    gives , so and .

  12. 12.

    Gym A costs $40 to join plus $25 a month; Gym B costs $10 plus $30 a month. After how many months do they cost the same?

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    gives , so months (both cost $190).
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MPhil in Mathematics, Founder of Mathovia
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Systems of Equations Practice Problems

These problems follow the systems of linear equations lesson. Each one checks one idea: what a solution is, how many solutions to expect, or how to find the solution when both equations are solved for .

Quick reference

SlopesInterceptsNumber of solutions
differentanythingone
the samedifferentnone
the samethe sameinfinitely many

Common mistakes to watch for

  • Choosing a slow method. Use substitution when a variable is already isolated, elimination when coefficients line up.
  • Not checking the solution. Substitute the pair into both original equations.
  • Sign errors when distributing. Bracket substituted expressions and distribute negatives carefully.
  • Misreading special cases. Parallel lines give no solution; identical lines give infinitely many.

Frequently Asked Questions

How do I check whether a point is a solution of a system?+

Substitute its coordinates into every equation. It is a solution only if every equation becomes true.

How can I tell how many solutions a system has without solving it?+

Compare slopes and intercepts. Different slopes give one solution, the same slope with different intercepts gives none, and identical lines give infinitely many.

How do I type my answer?+

Type an ordered pair such as (2, 5) or x = 2, y = 5. For counting questions type 0, 1 or infinitely many.

Calculators for this topic

System of Equations Calculator

More practice problems

Substitution Method Practice Problems Elimination Method Practice Problems Graphing Systems of Equations Practice Problems