Solving Systems of Linear Equations Using Substitution Method

Introduction

• Focus on solving systems of linear equations in two variables using the substitution method.
• Two examples will be used to illustrate the process.

Key Steps in Substitution Method

1. Solve for a Variable

   • Choose one equation and solve for one of the variables (x or y).
   • Substitute this expression into the other equation.

2. Substitution

   • Replace the chosen variable in the second equation with the expression obtained.
   • Ensure to use parentheses to group terms correctly.

3. Solve for the Remaining Variable

   • Solve the resulting equation to find the value of the second variable.

4. Back Substitution

   • Substitute the found value back into one of the original equations to solve for the first variable.

5. Coordinate Pair

   • Write the solution as a coordinate pair (x, y).

Example 1

• Given: x = 4 + y
• Substitute: Replace x in the second equation.
  • Equation: 8 + 2y - y = 7
• Solve for y: Simplify to find y = -1.
• Back Substitute for x:
  • x = 4 + (-1) = 3
• Solution as Coordinate Pair: (3, -1)
  • Interpretation: Graphically, lines intersect at (3, -1).

Example 2

• More Complex System
• Given Equations:
  • First: Not provided
  • Second: Solve for y in the form y = -1 - 3x by subtracting 3x from both sides.
• Substitute y in the First Equation:
  • Substitute y = -1 - 3x into the first equation.
  • Equation: 5x + 2(-1 - 3x) = -1
• Solve for x:
  • Distribute and combine like terms to find x = -1.
• Back Substitute for y:
  • y = -1 - 3(-1) = 2
• Solution as Coordinate Pair: (-1, 2)

Verification

• Substitute solutions back into both equations to verify they satisfy both.
• Ensure both equations are true with the found coordinate pair.

Next Steps

• Transition to learning the elimination method for solving systems of equations.
• Refer to the next video for elimination method insights.