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Systems by Elimination

A free Algebra I lesson from the “Systems of Linear Equations and Inequalities” unit, with a worked example and practice problems including step-by-step solutions.

Elimination solves a system by adding or subtracting equations so one variable disappears. The goal is to create opposite coefficients, such as 3x and -3x, then combine the equations to solve for the remaining variable. This method matters because it is efficient when both equations are in standard form or when substitution would create fractions. When practicing, line up like terms, decide which variable can be eliminated, and multiply one or both equations if needed. A common mistake is eliminating one variable correctly but forgetting to use the solution to find the second variable.

What you'll learn

Why it matters: Elimination is useful for comparing combined totals, such as ticket sales, orders, or mixtures, when one category can be canceled cleanly.

Worked example

Problem. Solve x + y = 9 and x - y = 3.

  1. Add the equations: 2x = 12.
  2. Divide by 2 to get x = 6.
  3. Substitute into x + y = 9, so y = 3.

Answer: (6, 3)

Practice problems

1. Solve x + y = 10 and x - y = 4. Enter x.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Solve x + y = 10 and x - y = 4. Enter x.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Add the equations.
  4. 2x = 14, so x = 7.
  5. Check the result by substituting or estimating: the response should match 7 and make sense in the original problem.

Answer: 7

2. For x + y = 10 and x - y = 4, enter y.

Show solution
  1. Warm-up: First identify exactly what the question is asking: For x + y = 10 and x - y = 4, enter y.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Use x = 7.
  4. 7 + y = 10, so y = 3.
  5. Check the result by substituting or estimating: the response should match 3 and make sense in the original problem.

Answer: 3

3. In 2x + y = 8 and 3x - y = 7, which variable cancels when you add?

Choices: y · x · both variables · neither variable

Show solution
  1. Warm-up: First identify exactly what the question is asking: In 2x + y = 8 and 3x - y = 7, which variable cancels when you add?
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. y and -y are opposites.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: y

4. Solve 2x + y = 11 and 3x - y = 14. Enter x.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Solve 2x + y = 11 and 3x - y = 14. Enter x.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Add the equations: 5x = 25.
  4. x = 5.
  5. Check the result by substituting or estimating: the response should match 5 and make sense in the original problem.

Answer: 5

5. For 2x + y = 11 and 3x - y = 14, enter y.

Show solution
  1. Core Practice: First identify exactly what the question is asking: For 2x + y = 11 and 3x - y = 14, enter y.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Use x = 5 in 2x + y = 11.
  4. 10 + y = 11, so y = 1.
  5. Check the result by substituting or estimating: the response should match 1 and make sense in the original problem.

Answer: 1

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