Lesson 76: Solving Linear Systems by Graphing
One way to solve a system of linear equations is by graphing. The solution to the system corresponds to the point(s) where the lines intersect.
Types of solutions:
- One solution: Lines intersect at a single point.
- No solution: Lines are parallel and never intersect.
- Infinitely many solutions: Lines coincide (are identical).
Steps to Solve by Graphing
- Rewrite each equation in slope-intercept form (y = mx + b).
- Graph both lines on the same coordinate plane.
- Identify the intersection point, if any.
- Check the solution by substituting the coordinates into both equations.
Examples
Example 1:
Solve by graphing:
y = 2x + 1
y = −x + 4
Graph both lines → They intersect at (1, 3) → Solution: (1, 3)
Example 2:
Solve by graphing:
y = 3x − 2
y = 3x + 1
Graph both lines → Lines are parallel → No solution
Example 3:
Solve by graphing:
y = −x + 2
y = −x + 2
Graph both lines → Lines coincide → Infinitely many solutions
Practice Problems
- Graph y = x + 1 and y = −2x + 4. Find the solution.
- Graph y = 3x − 5 and y = 3x + 2. Determine the type of solution.
- Graph y = 0.5x + 1 and y = −0.5x + 3. Find the intersection point.
- Graph y = 2x − 3 and y = 2x − 3. What is the solution?
- Graph y = −x + 0 and y = 2x + 1. Find the solution.