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Year 10

Finding where two straight lines meet

Solve simultaneous equations and interpret the answer as a point on both graphs.

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The idea

At an intersection, the same coordinates satisfy both equations. When each equation gives y in terms of x, set their right-hand sides equal to find the common x-coordinate, then substitute to find y.

Before you begin: Read coordinates and solve a two-step linear equation.

Worked example

Find the intersection of these two lines.

y=2⁢x+1,
y=−x+7
(2, 5)y = 2x + 1y = −x + 717270xy
Both equations give y = 5 when x = 2. The axes use the same scale.
Step 1

Both expressions represent the same y-value at the intersection.

2⁢x+1=−x+7
Step 2

Add x to both sides and subtract 1. Then divide by 3.

3⁢x=6
⇒x=2
Step 3

Substitute into the first equation to find y. Check in the second equation as well.

y=2⁢(2)+1=5,
y=−2+7=5
Step 4

State the answer as an ordered pair: horizontal coordinate first, vertical coordinate second.

(x,y)=(2,5)

A mistake to avoid

Finding x alone is not the intersection point. Find y as well, and check that the pair lies on both lines.

Your turn.

Find the intersection of this new pair.

y=3⁢x−2,
y=x+4
Show the solution
3⁢x−2=x+4
⇒x=3, y=7

The point is (3, 7). Substitution gives 7 for y in both equations.

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