1. **State the problem:** Solve the system of linear equations: $$4x + 2y = -16$$ $$5x + 9y = -85$$ 2. **Formula and method:** We will use the substitution or elimination method. Here, elimination is convenient. 3. **Eliminate one variable:** Multiply the first equation by 9 and the second by 2 to align coefficients of $y$: $$9(4x + 2y) = 9(-16) \Rightarrow 36x + 18y = -144$$ $$2(5x + 9y) = 2(-85) \Rightarrow 10x + 18y = -170$$ 4. **Subtract the second from the first to eliminate $y$:** $$ (36x + 18y) - (10x + 18y) = -144 - (-170) $$ $$ \Rightarrow (36x - 10x) + (18y - 18y) = -144 + 170 $$ $$ \Rightarrow 26x + \cancel{0} = 26 $$ 5. **Solve for $x$:** $$ 26x = 26 $$ $$ \Rightarrow x = \frac{26}{26} $$ $$ \Rightarrow x = 1 $$ 6. **Substitute $x=1$ into the first equation to find $y$:** $$4(1) + 2y = -16$$ $$4 + 2y = -16$$ $$2y = -16 - 4$$ $$2y = -20$$ $$y = \frac{-20}{2}$$ $$y = -10$$ 7. **Final answer:** $$\boxed{x=1, y=-10}$$ This means the solution to the system is the point $(1, -10)$ where both lines intersect.