1Cademy - Solving $$ \left\{4x + y = -5,\; -2x - 2y = -2\right\} $$ by Elimination

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Solving a System of Linear Equations by Elimination

Example

Solving $\left\{4x + y = -5,; -2x - 2y = -2\right\}$ by Elimination

Solve the system $\left\{\begin{array}{l} 4x + y = -5 -2x - 2y = -2 \end{array}\right.$ using the elimination method.

Step 1 — Write both equations in standard form. Both equations are already in the standard form $Ax + By = C$ .

Step 2 — Make the coefficients of one variable opposites. To eliminate $y$ , multiply the first equation by $2$ so that the $y$ -coefficients become $2$ and $-2$ : $2(4x + y) = 2(-5) \implies 8x + 2y = -10$

The system becomes: $\left\{\begin{array}{l} 8x + 2y = -10 -2x - 2y = -2 \end{array}\right.$

Step 3 — Add the equations to eliminate one variable. Adding the left and right sides together: $8x + 2y - 2x - 2y = -10 - 2$ $6x = -12$

The $y$ -terms cancel because $2y + (-2y) = 0$ .

Step 4 — Solve for the remaining variable. Divide both sides by $6$ : $x = -2$

Step 5 — Substitute back into an original equation. Substitute $x = -2$ into the first original equation $4x + y = -5$ : $4(-2) + y = -5$ $-8 + y = -5$ $y = 3$