How to Find the Intersection of Two Linear Equations

By Ainsley Patterson

With graphs, complex equations, and the many different shapes that can be involved, it is no wonder that math is one of the most dreaded subjects for many students. Let me guide you through one type of mathematical problem that you are likely to encounter sometime during your high school math career--how to find the intersection of two linear equations.

Start by knowing that your answer will be in the form of coordinates, meaning that your final answer should be in the form (x, y). This will help you remember that you need to solve not only for an x-value but also for a y-value.

Designate one equation as Line 1 and the other equation as Line 2 so that if you need to discuss this with a fellow student or a teacher you are able to keep the two linear equations straight.

Solve each equation so that they are both equations with the y variable on one side of the equation by itself and the x variable on the other side of the equation with all the functions and numbers. For example, the two equations below are in the format that your equations need to be in before you begin. Line 1: y = 3x+6 Line 2: y = -4x+9

Set the two equations equal to each other. For example, with the two equations from above: 3x+6 = -4x+9

Solve this new equation for x following the order of operations (parentheses, exponents, multiplication/division, addition/subtraction). For example, with the equation from above: 3x+6 = -4x+9 3x = -4x+3 (subtracting 6 from both sides) 0 = -7x+3 (subtracting 3x from both sides) -7x = -3 (subtracting 3 from both sides) x = 3/7 (divide both sides by -7)

Plug your value for x into either of the original equations and solve for y. For our equations from before: 3x+6 = y 3(3/7)+6 = y 9/7+6 = y 7 2/7 = y

Plug your value for x into the other equation to double check your y value. -4x+9 = y -4(3/7)+9 = y -12/7+9 = y 7 2/7 = y

Put your x and y values into coordinate form for your final answer. So, for our example our final answer would be (3/7, 7 2/7).

About the Author

Based in Ypsilanti, Mich., Ainsley Patterson has been a freelance writer since 2007. Her articles appear on various websites. She especially enjoys utilizing her more than 10 years of craft and sewing experience to write tutorials. Patterson is working on her bachelor's degree in liberal arts at the University of Michigan.