How to calculate points of intersection of straight lines

How to calculate points of intersection of straight lines

Two straight lines if they are nonparallel and do not coincide, are surely crossed in one point. To find coordinates of this place – means to calculate points of intersection of straight lines. Two crossed straight lines always lie in one plane therefore it is enough to consider of them in the Cartesian plane. Let's sort on an example how to find the general point of straight lines.


1. Take the equations of two straight lines, remembering that the straight line equation in the Cartesian system of coordinates the equation of a straight line looks as ah + vu + with =0, and a, v, s – usual numbers, and x and at – coordinates of points. For an example find points of intersection of straight lines of the 4th +3u-6=0 and the 2nd + at-4=0. For this purpose find decision systems of these two equations.

2. For the solution of a system of the equations change each of the equations so that y was faced by identical coefficient. As the coefficient before at is equal in one equation 1, just increase this equation by number 3 (coefficient before at in other equation). For this purpose increase each element of the equation by 3: (2kh*3)+ (u*3) - (4*3)= (0*3) also receive the usual equation of the 6th +3u-12=0. If coefficients before at were other than unit in both equations, it would be necessary to multiply both equalities.

3. Subtract another from one equation. For this purpose subtract one left part of another from the left part and in the same way treat right. Receive such expression: (the 4th +3u-6) - (the 6th +3u-12) =0-0. As the bracket is faced by the sign "-", change all signs in brackets for opposite. Receive such expression: the 4th +3u-6 - to a 6kh-3 +12=0. Simplify expression and you will see that the variable at disappeared. The new equation looks so: - the 2nd +6=0. Transfer number 6 to other member of equation, and from the turned-out equality - the 2nd =-6 express x: x = (-6) / (-2). Thus, you received also =3.

4. Substitute value x =3 in any equation, for example, in the second and receive such expression: (2*3) + at-4=0. Simplify and express at: at =4-6=-2.

5. Write down the received values x and at in the form of point coordinates (3;-2). These there will also be a solution of a task. Check the received value by a substitution method in both equations.

6. If straight lines are not given in the form of the equations, and given just on the plane, find coordinates of a point of intersection graphically. For this purpose prolong straight lines so that they were crossed, then lower on an axis oh and ou perpendiculars. Crossing of perpendiculars with axes oh and ou, will be coordinates of this point, look at the drawing and you will see that coordinates of a point of intersection x =3 and at =-2, that is a point (3;-2) also there is a solution of a task.

Author: «MirrorInfo» Dream Team