1Cademy - Solving $$\{x + y = -1,\; y = x + 5\}$$ by Substitution

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Solving a System of Linear Equations by Substitution

Example

Solving $\{x + y = -1,; y = x + 5\}$ by Substitution

Solve the system $\left\{\begin{array}{l} x + y = -1 y = x + 5 \end{array}\right.$ using the substitution method.

Because the second equation is already solved for $y$ , Step 1 of the substitution procedure is already complete — no algebraic rearrangement is needed before substituting.

Step 2 — Substitute into the other equation. Replace $y$ in the first equation with the expression $x + 5$ : $x + (x + 5) = -1$

Step 3 — Solve the resulting one-variable equation. Combine the like terms $x + x = 2x$ : $2x + 5 = -1$ Subtract $5$ from both sides: $2x = -6$ Divide both sides by $2$ : $x = -3$

Step 4 — Find the other variable. Substitute $x = -3$ into the second original equation $y = x + 5$ : $y = -3 + 5 = 2$

Step 5 — Write the solution as an ordered pair: $(-3, 2)$

Step 6 — Check in both original equations:

First equation: $-3 + 2 = -1$ . Since $-1 = -1$ is true ✓
Second equation: $2 = -3 + 5 = 2$ . Since $2 = 2$ is true ✓

Both equations are satisfied, confirming that $(-3, 2)$ is the solution of the system. This example illustrates the shortcut mentioned in the substitution method: when one equation is already solved for a variable, the process begins directly at Step 2, making the overall work shorter.

Updated 2026-06-29

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OpenStax Elementary Algebra

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