Method guideAlgebra IIntermediate9 min read

Solving systems by substitution

Replace one variable with an equal expression, solve the resulting one-variable equation, then recover and verify the second coordinate.

y=f(x),\quad g(x)+y=c\Rightarrow g(x)+f(x)=c

Method guide

How to recognize the method, run it, and know when it is the wrong choice.

Updated July 13, 2026

MethodStart here

Substitution is equality in action: if y equals an expression, that expression can replace y everywhere.

Isolate the cheaper variable

If neither variable is isolated, choose the coefficient ±1 when possible. Solving for that variable avoids introducing fractions.

Keep the entire replacement expression in parentheses, especially when it is multiplied or subtracted.

x=7-2y\Rightarrow3x-y=4\to3(7-2y)-y=4

After substitution, the equation contains one variable. Solve it normally, then use the simpler original equation to find the other value.

The result is an ordered pair. A lone x-value is only half of a system solution.

x=3\Rightarrow y=2(3)-1=5

If substitution produces a contradiction, the graphs never meet. If it produces an identity, the equations describe the same line.

Do not divide by an expression that could be zero just to make the variable reappear; interpret the identity or contradiction directly.

0=5\Rightarrow\varnothing\qquad0=0\Rightarrow\text{infinitely many}

Worked exampleReplace an equal quantity

2(3y-4)+y=13

Replace x in the second equation.

7y-8=13\Rightarrow7y=21\Rightarrow y=3

Solve for y.

x=3(3)-4=5

Back-substitute.

2(5)+3=13

Verify the pair.

Result

\boxed{(5,3)}

Watch for

Keep