Section 5.3: Graphing Systems
Graphing is a method for solving systems of linear equations by plotting each equation on the same coordinate plane. The solution is the point where the lines intersect.
Example 1
Solve the system by graphing:
\( y = 2x + 1 \)
\( y = -x + 4 \)
Step 1: Graph \( y = 2x + 1 \) → slope 2, y-intercept 1.
Step 2: Graph \( y = -x + 4 \) → slope -1, y-intercept 4.
Step 3: Find intersection point → \( x = 1, y = 3 \).
Solution: \( (x,y) = (1,3) \)
Example 2
Solve:
\( x + y = 5 \)
\( x - y = 1 \)
Step 1: Rewrite each equation in slope-intercept form:
\( y = -x + 5 \)
\( y = x - 1 \)
Step 2: Graph both lines. Intersection → \( x = 3, y = 2 \).
Solution: \( (x,y) = (3,2) \)
Practice Problems
- Graph \( y = 3x - 2 \) and \( y = -x + 4 \), find the solution.
- Graph \( 2x + y = 6 \) and \( x - y = 2 \), find the intersection point.
- Graph \( y = \frac{1}{2}x + 1 \) and \( y = -2x + 5 \), solve the system.
- Graph \( x + y = 7 \) and \( 3x - y = 5 \), determine the solution.
- Graph \( y = -x + 3 \) and \( y = x - 1 \), find the intersection.