Intersection of Two Lines

Idea

A point is on a line when it satisfies that line's equation. So a point on two lines must satisfy both equations at the same time.

That is all an intersection is. "Where do these lines cross?" and "what solves these two equations?" are the same question.

Rule

The intersection of $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$ is the solution of the two equations taken together. Solve by substitution or elimination; the pair $(x, y)$ you get is the meeting point.

\text{one solution} \iff \frac{a_1}{a_2} \ne \frac{b_1}{b_2}

When that condition fails the lines never cross once: they are parallel or the very same line.

How it is used

There are two methods. Learn both.

Method	Best when
Substitution	one equation is already solved for $x$ or $y$ , or easily can be
Elimination	the coefficients line up, so adding or subtracting kills a variable

Whichever you use, put your answer back into both equations. A crossing point must satisfy both of them. Checking only one proves nothing.

Three cases are worth spotting straight away.

A vertical line $x = c$ gives you $x$ at once. Put it into the other equation and you are finished in one step. A horizontal line $y = c$ works the same way.

If your working ends with something false, such as $0 = 5$ , there is no solution. The lines are parallel.

If it ends with $0 = 0$ , every point works. The two lines are the same line.

Worked example

Find where the lines $2x + y - 5 = 0$ and $x - y - 1 = 0$ cross. The $y$ terms are $+y$ and $-y$ , so adding the equations eliminates $y$ immediately:

(2x + y - 5) + (x - y - 1) = 3x - 6 = 0

That gives $x = 2$ . Substitute back into the second equation: $2 - y - 1 = 0$ , so $y = 1$ . Check both: $2(2) + 1 - 5 = 0$ and $2 - 1 - 1 = 0$ .

(2, 1)

Explore

Drag the ends of either line. The green circle marks where they meet, and the readout gives that point exactly, as a fraction when it is not a whole number. Now make the two slopes equal. The meeting point disappears.

Two lines, one plane

Try this. Use A(0,0), B(1,1), C(0,1), D(1,2) for parallel lines. Change D to (1,0) for perpendicular lines.

Point A · x-2Point A · y3Point B · x3Point B · y1Point C · x-3Point C · y-2Point D · x3Point D · y4

Blue slope: -0.4 · Green slope: 1 · Intersecting at (0.857, 1.857).