How to Find Where Two Lines Cross

This finds where $y = a_1x + b_1$ and $y = a_2x + b_2$ cross. At that point both lines give the same $y$, so setting the right-hand sides equal gives $x$.

$a_1$ and $a_2$ are the slopes, $b_1$ and $b_2$ the intercepts. Putting that $x$ back into either line gives $y$. The angle between the lines is also shown.

The angle

The angle comes from $\tan\theta = \left|\dfrac{a_1 - a_2}{1 + a_1a_2}\right|$. When $1 + a_1a_2$ is 0, meaning the slopes multiply to $-1$, the lines are perpendicular and the angle is 90 degrees.

Example

For $y = 2x + 1$ and $y = -x + 7$, setting $2x + 1 = -x + 7$ gives $3x = 6$, so $x = 2$. Putting that back into $y = 2x + 1$ gives $y = 5$. The angle follows from $\tan\theta = \left|3 \div (-1)\right| = 3$, about 71.6 degrees.

Notes

Two lines with the same slope are parallel and never cross. If the intercepts match as well they are the same line, and no single crossing point exists. Neither case is calculated.

A vertical line such as $x = 3$ cannot be written in this form, because it has no slope.