Solving Systems of Linear Equations Graphically

Introduction

A system of linear equations consists of two or more linear equations with the same variables. Solving a system of equations graphically means plotting each line on the Cartesian plane and finding the point where they intersect. This intersection point is the solution to the system.

In this lesson, we will learn how to graph each equation in a system and interpret the intersection point to find the solution.

Steps to Solve a System Graphically

  1. Write each equation in slope-intercept form: \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
  2. Plot the y-intercept (b) of each line on the Cartesian plane.
  3. Use the slope to plot a second point for each line, then draw the line through these points.
  4. Identify the point where the lines intersect. This point represents the solution to the system.

If the lines intersect at a point, the system has one solution. If the lines are parallel, there is no solution (the system is inconsistent). If the lines overlap, there are infinitely many solutions.

Practice Questions

  1. Solve the following system of equations graphically:
    • \( y = 2x + 1 \)
    • \( y = -x + 4 \)
    Solution

    1. For \( y = 2x + 1 \), plot the y-intercept at (0, 1) and use the slope of 2 to find another point (1, 3).

    2. For \( y = -x + 4 \), plot the y-intercept at (0, 4) and use the slope of -1 to find another point (1, 3).

    The lines intersect at (1, 3), so the solution is \( (1, 3) \).

  2. Solve the following system graphically:
    • \( y = -\frac{1}{2}x + 2 \)
    • \( y = \frac{1}{2}x - 1 \)
    Solution

    1. For \( y = -\frac{1}{2}x + 2 \), plot the y-intercept at (0, 2) and use the slope of -1/2 to plot a second point (2, 1).

    2. For \( y = \frac{1}{2}x - 1 \), plot the y-intercept at (0, -1) and use the slope of 1/2 to plot a second point (2, 0).

    The lines intersect at (2, 0), so the solution is \( (2, 0) \).

  3. Solve the following system graphically:
    • \( y = 3x + 2 \)
    • \( y = 3x - 4 \)
    Solution

    1. Both lines have the same slope (3), so they are parallel and will never intersect.

    This system has no solution.

Additional Practice

Try these problems on your own:

  1. Graph the following system and find the solution:
    • \( y = x + 2 \)
    • \( y = -x + 4 \)
  2. Graph the system and determine if it has a solution:
    • \( y = 2x - 3 \)
    • \( y = -2x + 5 \)
  3. Graph and solve the system below:
    • \( y = -x - 1 \)
    • \( y = x + 3 \)
→Applications of Linear Relations