Solving Systems of Linear Equations by Substitution

Substitution Method Overview

  • Substitution solves a system of linear equations by isolating one variable on one side of an equation and replacing that variable's expression in the second equation.
  • Parentheses must always be used when substituting expressions to ensure proper distribution and mathematical operations.

Step-by-Step System Solution

  • Given system of equations:
    • 3x+4y=53x + 4y = 5
    • y=2x+4y = 2x + 4
  • Substitute the equivalent expression 2x+42x + 4 for yy into the first equation:
    • 3x+4(2x+4)=53x + 4(2x + 4) = 5
  • Distribute and combine like terms to simplify to a single variable equation:
    • 3x+8x+16=53x + 8x + 16 = 5
    • 11x+16=511x + 16 = 5
  • Use inverse operations to solve for xx:
    • 11x=−1111x = -11
    • x=−1x = -1
  • Substitute x=−1x = -1 back into the isolated equation to solve for yy:
    • y=2(−1)+4y = 2(-1) + 4
    • y=−2+4y = -2 + 4
    • y=2y = 2

System Solution Form

  • A solution to a system of equations consists of both an xx and a yy value expressed as a single coordinate point.
  • The resulting solution point for this system is (−1,2)(-1, 2).