1Cademy - Solving $$\{2x + y = 7,\; x - 2y = 6\}$$ by Substitution

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Solutions of a System of Equations

Example

Solving $\{2x + y = 7,\; x - 2y = 6\}$ by Substitution

Solve the system $\left\{\begin{array}{l} 2x + y = 7 \\ x - 2y = 6 \end{array}\right.$ using the substitution method.

Step 1 — Solve one equation for one variable. The first equation is the easiest to solve for $y$ because its coefficient is $1$ . Subtract $2x$ from both sides:

$y = 7 - 2x$

Step 2 — Substitute into the other equation. Replace $y$ in the second equation with $7 - 2x$ :

$x - 2(7 - 2x) = 6$

Step 3 — Solve the resulting one-variable equation. Distribute $-2$ across the parentheses and combine like terms:

$x - 14 + 4x = 6$

$5x - 14 = 6$

Add $14$ to both sides: $5x = 20$ . Divide both sides by $5$ : $x = 4$ .

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

$2(4) + y = 7$

$8 + y = 7$

$y = -1$

Step 5 — Write the solution as an ordered pair: $(4, -1)$ .

Step 6 — Check in both original equations:

First equation: $2(4) + (-1) = 8 - 1 = 7$ . Since $7 = 7$ is true ✓
Second equation: $4 - 2(-1) = 4 + 2 = 6$ . Since $6 = 6$ is true ✓

Both equations are satisfied, confirming that $(4, -1)$ is the solution of the system. This is the same answer obtained by graphing the same system, demonstrating that the substitution method yields the exact solution through purely algebraic steps.

Updated 2026-04-21

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