1. **State the problem:** Solve the system of linear equations: $$-5x - 8y = 17$$ $$2x - 7y = -17$$ 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 5 to align coefficients of $x$: $$5 \times (2x - 7y) = 5 \times (-17)$$ $$10x - 35y = -85$$ 4. Multiply the first equation by 2: $$2 \times (-5x - 8y) = 2 \times 17$$ $$-10x - 16y = 34$$ 5. **Add the two new equations to eliminate $x$:** $$(-10x - 16y) + (10x - 35y) = 34 + (-85)$$ $$\cancel{-10x} - 16y + \cancel{10x} - 35y = -51$$ $$-51y = -51$$ 6. **Solve for $y$:** $$y = \frac{-51}{-51} = 1$$ 7. **Substitute $y=1$ into one original equation to find $x$:** Using the second equation: $$2x - 7(1) = -17$$ $$2x - 7 = -17$$ $$2x = -17 + 7$$ $$2x = -10$$ 8. **Solve for $x$:** $$x = \frac{\cancel{2}x}{\cancel{2}} = \frac{-10}{2} = -5$$ 9. **Final solution:** $$x = -5, \quad y = 1$$ This means the two lines intersect at the point $(-5, 1)$.