Section 2.6: Solving Systems by Graphing
Solving systems of linear equations by graphing involves plotting each equation on the same coordinate plane and identifying the point of intersection, if any.
Steps:
- Rewrite each equation in slope-intercept form (\(y = mx + b\)).
- Plot the y-intercept of each line on the graph.
- Use the slope to plot a second point for each line.
- Draw both lines and identify the intersection point.
- The intersection point \((x, y)\) is the solution. If the lines are parallel, there is no solution. If the lines overlap, there are infinitely many solutions.
Example 1
Solve by graphing: \[ \begin{cases} y = 2x + 1 \\ y = -x + 4 \end{cases} \]
Step 1: Graph \( y = 2x + 1 \) (y-intercept = 1, slope = 2). Step 2: Graph \( y = -x + 4 \) (y-intercept = 4, slope = -1). Step 3: Identify intersection: \( 2x + 1 = -x + 4 \) → \( 3x = 3 \) → \( x = 1 \), \( y = 3 \) Solution: (1, 3)
Example 2
Solve by graphing: \[ \begin{cases} y = x - 2 \\ y = x + 1 \end{cases} \]
Both lines have the same slope but different y-intercepts → parallel lines → no solution.
Practice Problems
- Solve by graphing: \( y = 3x + 2 \) and \( y = -x + 6 \)
- Solve by graphing: \( y = -2x + 1 \) and \( y = 0.5x - 2 \)
- Determine if there is a solution: \( y = x + 3 \) and \( y = x - 1 \)
- Solve by graphing: \( y = 4x - 5 \) and \( y = -x + 7 \)
- Solve by graphing: \( y = -x + 2 \) and \( y = 2x - 1 \)