Lines

Line Intersection

Solve two line equations at the same time and verify that the intersection point satisfies both lines.

Example

Solve two line equations and plot their crossing point.

highlighted = computed this step

Step 1 — Plot both lines

Plot both lines and their crossing point.

Coordinate plot\text{Coordinate plot}
Line intersection plotLines y = 2x + 1 and y = -x + 7 meet at I(2, 5).ABI

Step 2 — First line

Read the first line.

line A: y=2x+1\text{line A: } y= \hl{2} x + \hl{1}

Step 3 — Second line

Read the second line.

line B: y=−x+7\text{line B: } y= -x + \hl{7}

Step 4 — Set equations equal

Set the two y-values equal.

2x+1=-1x+7\hl{2} x + \hl{1} = \hl{-1} x + \hl{7}

Step 5 — Solve for x

Solve the equation for x.

3x=6,x=23 x = 6 , x= \hl{2}

Step 6 — Solve for y

Substitute x to find y.

y=2⋅2+1=5y= 2 \cdot 2 + 1 = \hl{5}

Step 7 — Intersection

State the intersection point.

I=(2,5)I=( \hl{2} , \hl{5} )
line-intersection At an intersection, both line equations have the same x and y.