How Do You Find the Intersection of Two Lines?

By Staff WriterLast Updated Mar 26, 2020 10:42:03 AM ET
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To find the intersection point of two lines, you must know both lines' equations. Once those are known, solve both equations for "x," then substitute the answer for "x" in either line's equation and solve for "y." The point (x,y) is the point where both lines intersect.

  1. Find the equations of both lines

    Depending on the situation, both equations might be given, or the equations might have to be calculated from other given information or from an illustration. Calculate the equations in either a point/slope form or a slope/intercept form. Either form works with this process.

  2. Set both equations to equal y

    Once the equations are known, rearrange the equations to get the "x" variables on one side of the equal sign and the "y" variables on the other. Solve the equations to the point where one "y" is on one side of the equation, and all the "x" variables are on the other.

  3. Set both equations to equal each other

    Since both equations now equal "y," they equal each other. Put one of them on each side of the equal sign. This allows solving for "x."

  4. Solve the equations for x

    Using mathematical processes, solve the equations for the value of "x."

  5. Substitute the numerical value of "x" in the original equations

    Once you have the value of "x," substitute that value for the "x" variables in one of the original equations. Either equation gives the same answer.

  6. Solve for y

    Simply solve the newly written equation, and obtain the value of "y."

  7. Record the intersection point

    The intersection point of the two lines is the point (x,y). Once both values are obtained, that is the answer.