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Algebra 101 - Unit 4

Lesson 21: Solving Systems by Substitution

The substitution method is used to solve a system of equations by replacing one variable with an equivalent expression from another equation.

Steps to Solve by Substitution

  1. Solve one equation for one variable in terms of the other.
  2. Substitute this expression into the other equation.
  3. Solve the resulting equation for the remaining variable.
  4. Substitute back to find the other variable.
  5. Check the solution in both original equations.

Example 1

System:

Solution using substitution:

  1. Substitute y = 2x + 3 into 3x + y = 15:

    2. 3x + (2x + 3) = 15 → 5x + 3 = 15

  2. Solve for x: 5x = 12 → x = 12/5
  3. Substitute x = 12/5 into y = 2x + 3:

    4. y = 2(12/5) + 3 = 24/5 + 15/5 = 39/5

  4. Solution: (x, y) = (12/5, 39/5)

Example 2

System:

Solution:

  1. From x + y = 5 → y = 5 - x
  2. Substitute into 2x - y = 4 → 2x - (5 - x) = 4 → 2x - 5 + x = 4 → 3x = 9 → x = 3
  3. Substitute x = 3 into y = 5 - x → y = 2
  4. Solution: (3,2)

Practice Problems

  1. Solve by substitution: y = x + 4, 2x + y = 10
  2. Solve by substitution: 3x + 2y = 12, y = x - 1
  3. Solve by substitution: x - y = 2, 2x + 3y = 12
  4. Solve by substitution: y = -2x + 5, x + y = 1

Lesson Summary