1. **State the problem:** Solve the system of linear equations: $$\begin{cases} 5x + 9y = -10 \\ 7x + 10y = -1 \end{cases}$$ 2. **Choose a method:** We will use the elimination method to solve for $x$ and $y$. 3. **Eliminate one variable:** Multiply the first equation by 7 and the second by 5 to align coefficients of $x$: $$\begin{cases} 7(5x + 9y) = 7(-10) \\ 5(7x + 10y) = 5(-1) \end{cases}$$ which gives $$\begin{cases} 35x + 63y = -70 \\ 35x + 50y = -5 \end{cases}$$ 4. **Subtract the second equation from the first:** $$ (35x + 63y) - (35x + 50y) = -70 - (-5) $$ which simplifies to $$ 35x - \cancel{35x} + 63y - 50y = -70 + 5 $$ $$ 13y = -65 $$ 5. **Solve for $y$:** $$ y = \frac{-65}{13} = -5 $$ 6. **Substitute $y = -5$ into the first original equation:** $$ 5x + 9(-5) = -10 $$ $$ 5x - 45 = -10 $$ 7. **Solve for $x$:** $$ 5x = -10 + 45 $$ $$ 5x = 35 $$ $$ x = \frac{35}{5} = 7 $$ **Final answer:** $$ x = 7, \quad y = -5 $$ This solution satisfies both equations in the system.