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Algebra 101 - Unit 4

Lesson 22: Solving Systems by Elimination

The elimination method solves a system of equations by adding or subtracting equations to eliminate one variable, making it possible to solve for the other variable.

Steps to Solve by Elimination

  1. Arrange both equations in standard form: Ax + By = C.
  2. Multiply one or both equations if necessary so the coefficients of one variable are equal (or opposites).
  3. Add or subtract the equations to eliminate that variable.
  4. Solve the resulting equation for the remaining variable.
  5. Substitute back to find the other variable.
  6. Check the solution in both original equations.

Example 1

System:

Solution using elimination:

  1. Add the two equations to eliminate y:

    2. (2x + 3y) + (4x - 3y) = 12 + 6 → 6x = 18 → x = 3

  2. Substitute x = 3 into 2x + 3y = 12:

    3. 2(3) + 3y = 12 → 6 + 3y = 12 → 3y = 6 → y = 2

  3. Solution: (x, y) = (3, 2)

Example 2

System:

Solution:

  1. Add the two equations to eliminate y:

    2. (3x + 2y) + (5x - 2y) = 16 + 14 → 8x = 30 → x = 15/4

  2. Substitute x = 15/4 into 3x + 2y = 16:

    3. 3(15/4) + 2y = 16 → 45/4 + 2y = 16 → 2y = 16 - 45/4 = 19/4 → y = 19/8

  3. Solution: (x, y) = (15/4, 19/8)

Practice Problems

  1. Solve by elimination: x + y = 7, 2x - y = 4
  2. Solve by elimination: 3x + 4y = 20, 5x - 4y = 0
  3. Solve by elimination: 2x + 5y = 11, 3x - 5y = 4
  4. Solve by elimination: x - 2y = 1, 4x + 2y = 14

Lesson Summary