Method guideCalculus I · Algebra IIntermediate

Solving systems by substitution

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

LaTeX article Updated July 13, 2026

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

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}