How to Solve Linear Systems by Graphing

By Rachel Morgan

Linear equations are equations that when graphed on a coordinate plane form straight lines. You can remember this by the fact that the word “line” is in the term linear. A linear system is made up of two equations which when graphed will form two lines. There are a number of ways to solve a system of linear equations, one of them being the graphing method. It is necessary to be familiar with the slope-intercept formula (y = mx + b) of graphing in order to use this method of finding solutions to linear systems. When both equations are graphed using the slope-intercept, you will be able to see whether the lines intersect on any point. If they do, you have found the solution. If the lines never cross, the equations in the system do not share a common solution. An example problem will be used in order to illustrate the steps: y = -x -1; -3y + 4x = 24.

Put the first equation into slope-intercept form. Slope-intercept form is written as y = mx + b, where the x stands for the slope of the line and the b represents the y-intercept. The slope of a line is its steepness, and the y-intercept is where the line crosses the vertical y axis on the coordinate plane. The first equation in this example system is already written in this form, with the -1 as the slope (it’s understood that –x has a coefficient of -1) and -1 as the y-intercept. The slope is written like a fraction: -1/1 and the y-intercept is an ordered pair: (0,-1).

Take the second equation and write it in slope-intercept form. This equation will take more time because it is not already in the correct format. The x term must be moved to the right side of the equation, and all terms must be divided by -3 in order to get the y by itself. The result is a slope of 4/3 and a y-intercept of (0, -8).-3y + 4x = 24-3y = -4x + 24y = -4/-3x + 24/-3y = 4/3x – 8

Graph the first equation on a coordinate plane using graphing paper and a ruler. Plot the y-intercept first, then use the slope to the find other points. With a slope of -1/1, you move down one point from the y-intercept of (0,-1) and to the right one. More points in the other direction can be found by reversing the order of movements: go up one and to the left one.

Graph the line of the second equation using the slope and y-intercept. Plot (0,-8) first and then use the slope to find other points on the lines. Using the slope 4/3, move up four points from (0,-8) and to the right three. To plot points in the other direction, go down four and to the left three.

The point of intersection is (3,-4). Verify this point by substituting the x and y values into both equations in the system to see if they make true statements. Both 3 and -4 work in the equations, so this gives you both the graph and algebraic equations to back up your answers. { (3,-4) } is the solution to the system.y = -x – 1-3y + 4x = 24-4 = -3 – 1-4 = -4-3(-4) + 4(3) = 2412 + 12 = 2424 = 24

About the Author

Rachel Morgan began her writing career in 2008 after previously working in her state's community college system. She focuses on health and fitness writing, in addition to blogging for small businesses. An alumna of the University of North Carolina, Morgan has a bachelor's degree in public health and has studied PR in the past.