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All courses Algebra I · A-REI.6 12 of 76
Solve systems of two linear equations exactly and approximately using algebraic and graphical methods.

Solve a system by substitution when one variable is already isolated

Problem
Solve the system by substitution: \(y~=~2x~+~1\), \(x~+~y~=~10\).
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Hint

Substitute 2x + 1 for y in the equation x + y = 10.

Substitution works by replacing one variable with an equivalent expression from the other equation.

Solution walkthrough

01

Substitute the isolated expression

\[y~=~2x~+~1;~x~+~y~=~10~->~x~+~(2x~+~1)~=~10\]

The first equation defines y, so replace y in the second equation with the entire equivalent expression 2x + 1.

02

Solve for x

\[x~+~2x~+~1~=~10~->~3x~=~9~->~x~=~3\]

Combine the x-terms, subtract 1 from both sides, and divide by 3.

03

Back-substitute for y

\[y~=~2(3)~+~1~=~7\]

Use x = 3 in the original isolated equation y = 2x + 1.

04

Check both equations

\[7~=~2(3)~+~1;~3~+~7~=~10\]

The first equality verifies y = 2x + 1 and the second verifies x + y = 10, so the ordered pair satisfies the full system.

05

State the ordered pair

\[(3,~7)\]

Coordinates are written as (x,y), so x = 3 and y = 7 give the solution (3,7).

+

Another way

  1. Isolate x = 10 - y from the second equation, substitute into y = 2x + 1, and solve to obtain the same pair.

!

Common mistake

Do not replace y with only 2x. The +1 is part of the equivalent expression and must stay inside the substitution.

Solution walkthrough video