1Cademy - Solving $$\left\{3x - 4y = -9,\; 5x + 3y = 14\right\}$$ by Elimination

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Solving a System of Linear Equations by Elimination

Example

Solving $\left\{3x - 4y = -9,\; 5x + 3y = 14\right\}$ by Elimination

To solve the system $\left\{\begin{array}{l} 3x - 4y = -9 \\ 5x + 3y = 14 \end{array}\right.$ using the elimination method, follow these steps:

Step 1 — Write both equations in standard form. Both equations are already in standard form.

Step 2 — Make the coefficients of one variable opposites. To eliminate $y$ , note that the $y$ -coefficients are $-4$ and $+3$ . Multiply the first equation by $3$ and the second equation by $4$ so that the $y$ -coefficients become $-12$ and $+12$ :

$3(3x - 4y) = 3(-9) \implies 9x - 12y = -27$

$4(5x + 3y) = 4(14) \implies 20x + 12y = 56$

The system is now $\left\{\begin{array}{l} 9x - 12y = -27 \\ 20x + 12y = 56 \end{array}\right.$

Step 3 — Add the equations to eliminate $y$ .

$9x - 12y + 20x + 12y = -27 + 56$

$29x = 29$

The $y$ -terms cancel out.

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

$x = 1$

Step 5 — Substitute back into an original equation. Substitute $x = 1$ into the first original equation $3x - 4y = -9$ :

$3(1) - 4y = -9$

$3 - 4y = -9$

$-4y = -12$

$y = 3$

Step 6 — Write the solution as an ordered pair: $(1, 3)$ .

Step 7 — Check in both original equations:

First equation: $3(1) - 4(3) = 3 - 12 = -9$ . Since $-9 = -9$ is true ✓
Second equation: $5(1) + 3(3) = 5 + 9 = 14$ . Since $14 = 14$ is true ✓

Both equations are satisfied, confirming that the solution is $(1, 3)$ .

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Updated 2026-05-07

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References

OpenStax Intermediate Algebra

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Intermediate Algebra @ OpenStax

Ch.4 Systems of Linear Equations - Intermediate Algebra @ OpenStax

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