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All courses Math I · A-REI.11 7 of 59
Solve equations of the form f(x)=g(x) approximately by finding graph/table intersections.

Find an exact intersection by solving f(x) = g(x)

Problem
Find the exact intersection of y = 2x + 1 and y = -x + 7.
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Hint

At the intersection, both equations have the same y-value. Set 2x + 1 equal to -x + 7.

Once you find x, substitute it back into either equation to find the matching y-value.

Solution walkthrough

01

Set the equations equal

\[2x~+~1~=~-x~+~7\]

The two expressions both equal y at the intersection point.

02

Solve for x

\[\begin{aligned} 3x~=~6 \\ x~=~2 \end{aligned}\]

Add x to both sides and subtract 1 to isolate x.

03

Find y

\[y~=~2(2)~+~1~=~5\]

Substitute x = 2 into either equation to get y = 5.

04

Write the intersection

\[(2,~5)\]

The exact intersection point is (2, 5).

+

Another way

  1. You can substitute x = 2 into y = -x + 7 instead and still get y = 5.

!

Common mistake

A common mistake is to stop after finding x = 2. The intersection must be written as an ordered pair, so you still need y.

Coordinate graph of y = 2x + 1 and y = -x + 7, with their exact intersection (2, 5) highlighted.
Exact intersection: (2, 5)