1. **State the problem:** Solve the system of linear equations: $$-7x + 5y = -17$$ $$-8x + 3y = 5$$ 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 3 and the second equation by 5 to align coefficients of $y$: $$3(-7x + 5y) = 3(-17) \Rightarrow -21x + 15y = -51$$ $$5(-8x + 3y) = 5(5) \Rightarrow -40x + 15y = 25$$ 4. **Subtract the two equations to eliminate $y$:** $$(-21x + 15y) - (-40x + 15y) = -51 - 25$$ $$-21x + 15y + 40x - 15y = -76$$ $$(-21x + 40x) + (15y - 15y) = -76$$ $$19x + 0 = -76$$ 5. **Solve for $x$:** $$19x = -76$$ $$x = \frac{-76}{19}$$ $$x = -4$$ 6. **Substitute $x = -4$ into the first original equation to find $y$:** $$-7(-4) + 5y = -17$$ $$28 + 5y = -17$$ 7. **Isolate $y$:** $$5y = -17 - 28$$ $$5y = -45$$ $$y = \frac{-45}{5}$$ $$y = -9$$ **Final answer:** $$x = -4, \quad y = -9$$