Overview
This lecture explains how to solve systems of two linear equations using the substitution method, with step-by-step guidance through two example problems.
Substitution Method Steps
- Identify or rearrange one equation so a variable (x or y) is isolated.
- Substitute the isolated variable's expression into the other equation, replacing the variable.
- Solve the resulting single-variable equation.
- Substitute the found value back into either original equation to solve for the second variable.
- Write the solution as an ordered pair (x, y), representing the intersection point of the two lines.
- Optional: Check your solution by plugging values into both original equations to verify.
Example 1: Easy Substitution
- If the equation is already isolated (e.g., x = 4 + y), substitute into the second equation.
- Replace x in the second equation with (4 + y), using parentheses.
- Simplify and solve for y.
- Substitute the y-value back into the isolated equation to solve for x.
- Solution format: (3, -1), meaning x = 3 and y = -1.
Example 2: Rearrangement Needed
- Rearrange one equation to isolate y (e.g., y = -1 - 3x) for easier substitution.
- Substitute y's expression into the other equation, again using parentheses.
- Distribute, simplify, and solve for x.
- Substitute the x-value back into the isolated equation to solve for y.
- Solution format: (-1, 2), meaning x = -1 and y = 2.
Checking Solutions
- Substitute both (x, y) values into original equations to ensure both are satisfied.
Key Terms & Definitions
- System of Equations — A set of two or more equations involving the same variables.
- Substitution Method — A way to solve systems by isolating a variable in one equation and substituting into the other.
- Ordered Pair — A pair (x, y) representing the solution to a system, or where two graphs intersect.
Action Items / Next Steps
- Practice solving systems using both substitution and elimination methods.
- Watch the next video on the elimination method for solving systems.