1. We are given the system of linear equations: $$8x - 5y = 66$$ $$3x + 2y = 17$$ 2. Our goal is to find the values of $x$ and $y$ that satisfy both equations simultaneously. 3. One common method to solve such systems is the elimination method. We will eliminate one variable by making the coefficients of $y$ in both equations opposites. 4. Multiply the first equation by 2 and the second equation by 5 to align the coefficients of $y$: $$2(8x - 5y) = 2(66) \Rightarrow 16x - 10y = 132$$ $$5(3x + 2y) = 5(17) \Rightarrow 15x + 10y = 85$$ 5. Add the two new equations to eliminate $y$: $$16x - 10y + 15x + 10y = 132 + 85$$ $$31x = 217$$ 6. Solve for $x$: $$x = \frac{217}{31} = 7$$ 7. Substitute $x=7$ back into one of the original equations, for example, the second: $$3(7) + 2y = 17$$ $$21 + 2y = 17$$ 8. Solve for $y$: $$2y = 17 - 21 = -4$$ $$y = \frac{-4}{2} = -2$$ 9. The solution to the system is: $$x = 7, \quad y = -2$$