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All courses Algebra I · A-REI.11 7 of 76
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 \(y~=~-x~+~7\).
Your answer
Show answer choicesHide answer choices Work through the mathematics first, then compare your reasoning.Select the answer that matches your work.
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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

Translate intersection into equality

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

At an intersection, the two equations have the same input x and the same output y. Therefore their expressions for y must be equal.

02

Solve for the shared input

\[2x+1=-x+7~->~3x+1=7~->~3x=6~->~x=2\]

Adding x to both sides and then subtracting 1 preserves equality. Dividing by 3 gives the intersection's x-coordinate.

03

Find the shared output

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

Substitute the solved input 2 into the first source equation. The corresponding y-coordinate is 5.

04

Check both equations

\[2(2)+1=5;~-(2)+7=5\]

Both source rules produce 5 at x = 2, confirming that the point lies on both lines. The inspected answer graph also marks this crossing within x-bounds -1 to 5 and y-bounds -2 to 10.

05

Write the exact intersection

\[(2,~5)\]

An intersection is an ordered pair in (x,y) order, so the exact answer is choice A.

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Another way

  1. Graph both source lines and read their crossing, then verify the coordinates algebraically in both equations.

!

Common mistake

Do not stop at x = 2 or reverse the coordinates. The problem asks for an intersection point, so find y and write (x,y).

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

Solution walkthrough video