1Cademy - Solving $$\{3x - 2y = -2,\; 5x - 6y = 10\}$$ by Elimination

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Solving a System of Linear Equations by Elimination

Example

Solving $\{3x - 2y = -2,\; 5x - 6y = 10\}$ by Elimination

Solve the system $\left\{\begin{array}{l} 3x - 2y = -2 \\ 5x - 6y = 10 \end{array}\right.$ using the elimination method.

Step 1 — Write both equations in standard form. Both equations are already in the form $Ax + By = C$ , so no rewriting is needed.

Step 2 — Make the coefficients of one variable opposites. Neither pair of coefficients are already opposites. To eliminate $y$ , note that the $y$ -coefficients are $-2$ and $-6$ . Multiplying the first equation by $-3$ transforms its $y$ -coefficient from $-2$ to $+6$ , which is the opposite of $-6$ :

$-3(3x - 2y) = -3(-2) \implies -9x + 6y = 6$

The system becomes $\left\{\begin{array}{l} -9x + 6y = 6 \\ 5x - 6y = 10 \end{array}\right.$

Step 3 — Add the equations to eliminate $y$ . Adding the left sides and right sides:

$-9x + 6y + 5x - 6y = 6 + 10$

$-4x = 16$

The $y$ -terms cancel because $6y + (-6y) = 0$ .

Step 4 — Solve for the remaining variable. Divide both sides by $-4$ :

$x = -4$

Step 5 — Substitute back into an original equation. Substitute $x = -4$ into the first equation $3x - 2y = -2$ :

$3(-4) - 2y = -2$

$-12 - 2y = -2$

Add $12$ to both sides: $-2y = 10$ . Divide both sides by $-2$ : $y = -5$ .

Step 6 — Write the solution as an ordered pair: $(-4, -5)$ .

Step 7 — Check in both original equations:

First equation: $3(-4) - 2(-5) = -12 + 10 = -2$ . Since $-2 = -2$ is true ✓
Second equation: $5(-4) - 6(-5) = -20 + 30 = 10$ . Since $10 = 10$ is true ✓

Both equations are satisfied, confirming that $(-4, -5)$ is the solution. Unlike the previous elimination example — where only a positive multiplier was needed because one coefficient was already $+1$ — this system requires multiplying the first equation by $-3$ to create opposite $y$ -coefficients. This illustrates the strategy of choosing a negative multiplier so that the resulting coefficient has the opposite sign of the corresponding coefficient in the other equation.

Updated 2026-04-21

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OpenStax Elementary Algebra

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