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Math Practice

Substitution Method Practice Problems

Isolate one variable, substitute it into the other equation, solve, then back-substitute. The first problems already have a variable alone; later ones make you choose which variable to isolate and recognize the special cases.

  1. 1.

    Solve by substitution: and .

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    Substitute: , so and . Then .

  2. 2.

    Solve by substitution: and .

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    gives , so .

  3. 3.

    Solve by substitution: and .

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    gives , so and .

  4. 4.

    Solve by substitution: and .

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    gives , so and .

  5. 5.

    Solve by substitution: and .

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    gives , so and .

  6. 6.

    Solve by substitution: and .

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    Isolate . Then gives , so and .

  7. 7.

    Solve by substitution: and .

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    Isolate . Then gives , so and .

  8. 8.

    Solve by substitution: and .

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    Isolate . Then gives , so and .

  9. 9.

    Solve by substitution: and .

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    gives . The variable cancels and leaves a true statement, so there are infinitely many solutions.

  10. 10.

    Solve by substitution: and .

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    gives . A false statement means the lines are parallel: no solution.

  11. 11.

    Solve by substitution: and .

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    Isolate from the second equation. Then gives , so and .

  12. 12.

    A concert sold 200 tickets for $2,600. Adult tickets cost $14 and child tickets $9. How many adult tickets were sold?

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    and . Substitute : , so and .
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MPhil in Mathematics, Founder of Mathovia
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Substitution Method Practice Problems

These problems follow the substitution method lesson. Work them like a substitution method worksheet: write each step, and keep brackets around the expression you substitute.

The steps

  1. Isolate one variable in one equation.
  2. Substitute that expression, in brackets, into the other equation.
  3. Solve the one-variable equation.
  4. Back-substitute to find the other variable.
  5. Check the pair in both original equations.

Common mistakes to watch for

  • Substituting back into the same equation. After solving one equation for a variable, substitute into the other equation.
  • Dropping brackets. Substituting into gives ; the minus applies to both terms.
  • Stopping after one variable. Use the first value to find the second, and give the solution as an ordered pair.
  • Misreading special results. means no solution; means infinitely many.

Frequently Asked Questions

Which variable should I isolate?+

One with a coefficient of 1 or −1, so no fractions appear. If a variable is already alone, use that equation.

What if the variable disappears?+

A false statement such as −4 = 6 means no solution. A true statement such as 12 = 12 means infinitely many solutions.

How do I type my answer?+

Type an ordered pair such as (2, 5) or x = 2, y = 5, or type no solution or infinitely many.

Calculators for this topic

Substitution Method Calculator

More practice problems

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