Lesson 20: Solving Systems by Graphing
A system of equations is a set of two or more equations with the same variables. Solving the system means finding values of the variables that satisfy all equations simultaneously.
Solving by Graphing
Steps to solve a system by graphing:
- Graph each equation on the same coordinate plane.
- Identify the point where the lines intersect.
- The intersection point (x, y) is the solution to the system.
If the lines are parallel, there is no solution. If the lines coincide (overlap), there are infinitely many solutions.
Example: Solve the system by graphing
System:
- y = 2x + 1
- y = -x + 4
Steps:
- Graph y = 2x + 1 (slope 2, y-intercept 1)
- Graph y = -x + 4 (slope -1, y-intercept 4)
- Find the intersection point
The lines intersect at (1,3), so the solution is x = 1, y = 3.
Practice Problems
- Solve by graphing: y = x + 2 and y = -2x + 5
- Solve by graphing: y = 1/2x - 1 and y = 1/2x + 3
- Solve by graphing: y = -x + 1 and y = 2x - 4
- Graph y = 3x + 1 and y = 3x - 2 and describe the solution.
Lesson Summary
- A system of equations has a solution where the lines intersect.
- Parallel lines → no solution, coinciding lines → infinitely many solutions.
- Graphing gives a visual method to find solutions.