How to Find the Intersection of Three Lines

By Emily Stokes
Jupiterimages/BananaStock/Getty Images

Finding the intersection of three lines is not much different from finding the intersection of two lines, and both are challenges you will likely face in geometry study. Being able to find these intersections can save you time, and can also be a requirement of your teacher or professor on your next exam. This skill can also have real-life value in carpentry, graph making or other situations requiring a technical grasp of line and space.

Using the equations for each line, isolate the y value on one side of each equation and then set the equations equal to one another. For example, using y = (-x) + 2, y = x + 1 and y = (-1/2)x + 1, you can say that (-x) + 2 = x + 1 = (-1/2)x + 1. As you simplify this equation, remember that the left side value is from equation #1, the middle value is from equation #2 and the right side is from equation #3.

Using this three-sided equation, perform operations to eliminate any extra numbers or x-values possible --- and be sure to perform the same operation on each of the three sections of the equation. In the example, you can add x to each side of the equation to simplify to: 2 = 2x + 1 = (1/2)x + 1. Then you can subtract 1 from each side to simplify to: 1 = 2x = (1/2)x.

Determine all possible values for x after the equation is fully simplified. This can be done by breaking the equation down into pieces and solving for x. For example, the previous equation can be broken into 1 = 2x, 1 = (1/2)x and 2x = (1/2)x, which in each situation would require that x = 1/2, x = 2 and x = 0, respectively. These are the x-value coordinates for your intersection points (x,y).

Solve the equations for the y coordinate value by plugging appropriate x values into appropriate equations; this is where it is important to remember which part of your equations originated from which one of the three separate initial equations. For example, the simplified three-part equation 1 = 2x = (1/2)x is a combination of equation #1 = equation #2 = equation #3. When breaking this simplified equation into three separate parts to find the possible values for x, you were also doing this: equation #1 = equation #2, equation #1 = equation #3 and equation #2 = equation #3. In the example, the discovered x value for equation #1 = equation #2 was x = (1/2); therefore, you must plug this value into the original equations of either equation #1 or equation #2. This value cannot be plugged into the original equation #3, because the two equations used to find this x value were #1 and #2.

Format your plot points in (x,y) format after you find each y-value. The example answers would look like this: (0,1), (2,0) and (1/2,3/2).

About the Author

Emily Stokes graduated from Dickinson College with a B.A. in English and is pursuing graduate studies in writing at Sarah Lawrence College. Stokes has worked as a writing tutor at the Eberly Writing Center in Carlisle, Pa., and as a subcontracting writer for SKY University, LLC.