Solving Systems of Linear Equations using Substitution Method

Introduction

• Learning to solve systems of linear equations in two variables using substitution.
• Two examples will be covered.

Key Steps in Substitution Method

1. Choose a Variable to Solve For: Decide to solve for either x or y.
2. Substitute the Expression: Once a variable is isolated, substitute it into the other equation to form a single-variable equation.
3. Solve the Single-Variable Equation: Solve for the variable that remains.
4. Substitute Back to Find the Other Variable: Use the found value to determine the value of the other variable.
5. Write the Solution as a Coordinate Pair: Express the solution as (x, y).
6. Check the Solution: Substitute back to ensure that the solution satisfies both equations.

Example 1

• Given: x = 4 + y
• Substitute x in the second equation: Replace x with (4 + y).
• Resulting equation: 8 + 2y - y = 7
• Simplify and solve for y:
  • 2y - y = y
  • 8 + y = 7
  • y = -1
• Substitute y = -1 back to find x:
  • x = 4 + (-1)
  • x = 3
• Solution: (3, -1)
• Interpretation: The lines intersect at (3, -1).

Example 2

• More challenging as no variable is isolated initially.
• Choose to solve for y in the equation: y = -1 - 3x
• Substitute y in the first equation: 5x + 2(-1 - 3x) = -1
• Simplify and solve for x:
  • Distribute 2: -2 - 6x
  • Combine like terms: 5x - 6x = -x
  • -x + 2 = -1
  • x = -1
• Substitute x = -1 back to find y:
  • y = -1 - 3(-1)
  • y = -1 + 3
  • y = 2
• Solution: (-1, 2)
• Interpretation: The lines intersect at (-1, 2).

Verification

• Check solutions by substituting back into original equations.
• Ensure both solutions satisfy each original equation.

Next Steps

• Elimination Method: Follow the next video to learn solving systems using the elimination method.