Solve a system of equations by graphing
A system of equations is a set of two or more equations. The solution is the one point (x, y) that makes ALL the equations true. Graphing is a visual way to find this solution—it's the point where the lines intersect!
Concept
To solve a system of linear equations by graphing, you graph each line on the same coordinate plane and find the point where they cross. **The Solution:** The solution to a system of linear equations is the **point of intersection**. This (x, y) coordinate is the only point that lies on both lines, meaning it is the only point that satisfies both equations. **Steps to Solve by Graphing:** 1. **Write both equations in slope-intercept form (y = mx + b).** This makes them easy to graph. 2. **Graph the first equation.** * Start by plotting the y-intercept (the 'b' value) on the y-axis. * From the y-intercept, use the slope (m = rise/run) to find a second point. * Draw a straight line through these two points. 3. **Graph the second equation** on the same coordinate plane using the same method. 4. **Find the point of intersection.** This is the (x, y) coordinate where the two lines cross. 5. **Check your solution.** Plug the x and y values from the intersection point into both of the original equations to make sure they are both true. **Example: Solve the system y = 2x - 1 and y = -x + 5** 1. **Slope-intercept form:** Both equations are already in this form. 2. **Graph y = 2x - 1:** * Y-intercept (b) is -1. Plot a point at (0, -1). * Slope (m) is 2, or 2/1. From (0, -1), rise 2 and run 1 to find a second point at (1, 1). * Draw the line. 3. **Graph y = -x + 5:** * Y-intercept (b) is 5. Plot a point at (0, 5). * Slope (m) is -1, or -1/1. From (0, 5), go down 1 and right 1 to find a second point at (1, 4). * Draw the line. 4. **Find intersection:** The lines cross at the point **(2, 3)**. 5. **Check:** * In y = 2x - 1: Does 3 = 2(2) - 1? 3 = 4 - 1. Yes. * In y = -x + 5: Does 3 = -(2) + 5? 3 = 3. Yes. * The solution is correct. **Types of Solutions:** * **One Solution:** The lines intersect at one point (most common). * **No Solution:** The lines are parallel and never intersect. They have the same slope but different y-intercepts. * **Infinitely Many Solutions:** The two equations represent the exact same line. They have the same slope and the same y-intercept.
Try it
Practice: Solve a system of equations by graphing.