Solve System Eb1A7B

1. **State the problem:** Solve the system of linear equations: $$\begin{cases} 14x + 5y = 31 \\ 2x - 3y = -29 \end{cases}$$ 2. **Choose a method:** We will use the elimination method to solve for $x$ and $y$. 3. **Eliminate one variable:** Multiply the second equation by 7 to align the $x$ coefficients: $$7 \times (2x - 3y) = 7 \times (-29)$$ $$14x - 21y = -203$$ 4. **Subtract the first equation from this new equation:** $$\begin{aligned} &(14x - 21y) - (14x + 5y) = -203 - 31 \\ &14x - 21y - 14x - 5y = -234 \\ &-26y = -234 \end{aligned}$$ 5. **Solve for $y$:** $$y = \frac{-234}{-26} = \frac{234}{26}$$ Simplify the fraction by dividing numerator and denominator by 2: $$y = \frac{\cancel{234}{}^{\div 2}}{\cancel{26}{}^{\div 2}} = \frac{117}{13}$$ 6. **Substitute $y$ back into one of the original equations to find $x$:** Use the second equation: $$2x - 3y = -29$$ Substitute $y = \frac{117}{13}$: $$2x - 3 \times \frac{117}{13} = -29$$ $$2x - \frac{351}{13} = -29$$ 7. **Isolate $2x$:** $$2x = -29 + \frac{351}{13}$$ Convert $-29$ to a fraction with denominator 13: $$-29 = \frac{-377}{13}$$ So: $$2x = \frac{-377}{13} + \frac{351}{13} = \frac{-377 + 351}{13} = \frac{-26}{13}$$ 8. **Solve for $x$:** $$x = \frac{\frac{-26}{13}}{2} = \frac{-26}{13} \times \frac{1}{2} = \frac{-26}{26} = -1$$ 9. **Final solution:** $$x = -1, \quad y = \frac{117}{13}$$