System Equations A30Fa8

1. **State the problem:** Solve the system of linear equations: $$10x - 2y = 44$$ $$-6x - 9y = 10$$ 2. **Choose a method:** We will use the elimination method to solve for $x$ and $y$. 3. **Make coefficients of $y$ equal:** Multiply the first equation by $9$ and the second equation by $2$ to align the coefficients of $y$: $$9(10x - 2y) = 9(44) \Rightarrow 90x - 18y = 396$$ $$2(-6x - 9y) = 2(10) \Rightarrow -12x - 18y = 20$$ 4. **Subtract the second equation from the first:** $$ (90x - 18y) - (-12x - 18y) = 396 - 20 $$ $$ 90x - 18y + 12x + 18y = 376 $$ $$ (90x + 12x) + (-18y + 18y) = 376 $$ $$ 102x + \cancel{-18y + 18y} = 376 $$ $$ 102x = 376 $$ 5. **Solve for $x$:** $$ x = \frac{376}{102} $$ Simplify the fraction by dividing numerator and denominator by 2: $$ x = \frac{\cancel{376}^ {188}}{\cancel{102}^{51}} $$ 6. **Substitute $x = \frac{188}{51}$ into the first original equation:** $$ 10x - 2y = 44 $$ $$ 10 \times \frac{188}{51} - 2y = 44 $$ $$ \frac{1880}{51} - 2y = 44 $$ 7. **Isolate $y$:** $$ -2y = 44 - \frac{1880}{51} $$ Convert $44$ to fraction with denominator $51$: $$ 44 = \frac{2244}{51} $$ $$ -2y = \frac{2244}{51} - \frac{1880}{51} = \frac{364}{51} $$ 8. **Solve for $y$:** $$ y = -\frac{1}{2} \times \frac{364}{51} = -\frac{364}{102} $$ Simplify by dividing numerator and denominator by 2: $$ y = -\frac{\cancel{364}^{182}}{\cancel{102}^{51}} $$ **Final solution:** $$ x = \frac{188}{51}, \quad y = -\frac{182}{51} $$