Lines

Line Intersection

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

line-intersection At an intersection, both line equations have the same x and y.

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} )