1Cademy - Solving $$\{x - 2y = -2,\; 3x + 2y = 34\}$$ by Substitution

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Solving a System of Linear Equations by Substitution

Example

Solving $\{x - 2y = -2,\; 3x + 2y = 34\}$ by Substitution

Solve the system $\left\{\begin{array}{l} x - 2y = -2 \\ 3x + 2y = 34 \end{array}\right.$ using the substitution method.

Step 1 — Solve one equation for one variable. In the first equation, $x$ has a coefficient of $1$ , making it the simplest variable to isolate. Add $2y$ to both sides:

$x = 2y - 2$

Step 2 — Substitute into the other equation. Replace $x$ in the second equation with $2y - 2$ :

$3(2y - 2) + 2y = 34$

Step 3 — Solve the resulting one-variable equation. Distribute $3$ across the parentheses:

$6y - 6 + 2y = 34$

Combine the like terms $6y + 2y = 8y$ :

$8y - 6 = 34$

Add $6$ to both sides: $8y = 40$ . Divide both sides by $8$ : $y = 5$ .

Step 4 — Find the other variable. Substitute $y = 5$ into the first original equation $x - 2y = -2$ :

$x - 2(5) = -2$

$x - 10 = -2$

$x = 8$

Step 5 — Write the solution as an ordered pair: $(8, 5)$ .

Step 6 — Check in both original equations:

First equation: $8 - 2(5) = 8 - 10 = -2$ . Since $-2 = -2$ is true ✓
Second equation: $3(8) + 2(5) = 24 + 10 = 34$ . Since $34 = 34$ is true ✓

Both equations are satisfied, confirming that $(8, 5)$ is the solution of the system. Unlike previous examples that isolated $y$ in Step 1, this example isolates $x$ instead — illustrating that the substitution method works equally well when solving for either variable, and the best choice is whichever variable already has a coefficient of $1$ .

Updated 2026-04-21

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OpenStax Elementary Algebra

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