Overview
This lecture explains how to solve systems of two linear equations using the elimination (addition) and substitution methods, with step-by-step examples.
Elimination (Addition) Method
- To use elimination, align equations and add/subtract them to eliminate one variable.
- Example: Given 2x + 3y = 8 and 5x - 3y = -1, add the equations to eliminate y.
- Result: 7x = 7; solve for x to get x = 1.
- Substitute x = 1 into either equation to solve for y, yielding y = 2.
- The solution is written as the ordered pair (1, 2).
- For equations where variables do not cancel directly, multiply one equation to create opposites.
- Example: For 2x + 5y = 19 and x - 2y = -4, multiply the second equation by -2 and add.
- Solve resulting equations to find x = 2 and y = 3, so the solution is (2, 3).
Substitution Method
- In substitution, solve one equation for one variable in terms of the other.
- Substitute this expression into the second equation to solve for the remaining variable.
- Example: With y = 5 - 2x and 4x + 3y = 13, substitute y into the second equation.
- After simplifying and solving, x = 1; then substitute x into the first equation to get y = 3.
- The solution is (1, 3).
- If both equations are solved for y, set them equal and solve for x.
- Example: If y = 3x + 2 and y = 7x - 6, set 3x + 2 = 7x - 6, solve for x, then for y; answer is (2, 8).
- For forms like 4x + 2y = 14 and 3x - 5y = -22, isolate y in the first equation, substitute into the second, and solve for x and y; answer is (1, 5).
Key Terms & Definitions
- System of equations — A set of two or more equations with the same variables.
- Elimination (Addition) Method — Solving by adding or subtracting equations to eliminate a variable.
- Substitution Method — Solving by replacing one variable with an equivalent expression from another equation.
- Ordered pair — A solution written as (x, y) that satisfies both equations.
Action Items / Next Steps
- Practice solving systems of equations using both elimination and substitution methods.
- Review class examples and attempt similar problems for mastery.