How to Solve a System of Equations by Substitution

Problem

y = 2x + 1,\quad 3x + y = 11

Answer

x = 2,\ y = 5

Substitution works well when one equation already gives a variable by itself. Replace that variable in the other equation, solve, then find the second value.

Step-by-step solution

Substitute 2x + 1 for y in the second equation.
$3x + (2x + 1) = 11$
Why: Both equations describe the same y, so the expressions are equal.
Combine like terms.
$5x + 1 = 11$
Why: 3x plus 2x is 5x.
Subtract 1 from both sides.
$5x = 10$
Why: Subtracting undoes the addition.
Divide both sides by 5.
$x = 2$
Why: Dividing by the coefficient leaves x alone.
Substitute x = 2 into the first equation to find y.
$y = 2(2) + 1 = 5$
Why: The solution needs both values.

Common mistakes

Substituting into the same equation. Use the other equation.
Dropping the parentheses around 2x + 1.
Stopping after finding x. The solution is the pair (2, 5).

Practice problems

Use the same method. Work each problem on paper, then open the answer to check.

$y = x + 2,\ x + y = 8$

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$x = 3,\ y = 5$
$y = 2x,\ x + y = 9$

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$x = 3,\ y = 6$
$y = x - 1,\ 4x + y = 14$

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$x = 3,\ y = 2$
$y = 2x - 3,\ x + y = 9$

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$x = 4,\ y = 5$
$y = 5x,\ 2x + y = 21$

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$x = 3,\ y = 15$