1Cademy - Solving $$\{4x - 3y = 9,\; 7x + 2y = -6\}$$ by Elimination

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Solving a System of Linear Equations by Elimination

Example

Solving $\{4x - 3y = 9,\; 7x + 2y = -6\}$ by Elimination

Solve the system $\left\{\begin{array}{l} 4x - 3y = 9 \\ 7x + 2y = -6 \end{array}\right.$ 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. In this system, no single multiplier applied to just one equation can produce opposite coefficients for either variable. The strategy is to multiply both equations by carefully chosen constants. To eliminate $y$ , note that the $y$ -coefficients are $-3$ and $+2$ . Multiply the first equation by $2$ and the second equation by $3$ so that the $y$ -coefficients become $-6$ and $+6$ :

$2(4x - 3y) = 2(9) \implies 8x - 6y = 18$

$3(7x + 2y) = 3(-6) \implies 21x + 6y = -18$

The system becomes $\left\{\begin{array}{l} 8x - 6y = 18 \\ 21x + 6y = -18 \end{array}\right.$

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

$8x - 6y + 21x + 6y = 18 + (-18)$

$29x = 0$

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

Step 4 — Solve for the remaining variable. Divide both sides by $29$ :

$x = 0$

Step 5 — Substitute back into an original equation. Substitute $x = 0$ into the second equation $7x + 2y = -6$ :

$7(0) + 2y = -6$

$2y = -6$

Divide both sides by $2$ : $y = -3$ .

Step 6 — Write the solution as an ordered pair: $(0, -3)$ .

Step 7 — Check in both original equations:

First equation: $4(0) - 3(-3) = 0 + 9 = 9$ . Since $9 = 9$ is true ✓
Second equation: $7(0) + 2(-3) = 0 - 6 = -6$ . Since $-6 = -6$ is true ✓

Both equations are satisfied, confirming that $(0, -3)$ is the solution. Unlike the previous elimination examples — where multiplying only one equation by a single constant was sufficient to create opposite coefficients — this system requires multiplying both equations by different constants. The choice of multipliers $2$ and $3$ produces opposite $y$ -coefficients ( $-6$ and $+6$ ), but other multiplier pairs would also work and yield the same solution.

Updated 2026-05-07

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