1. **State the problem:** Solve the system of linear equations: $$-9x + 4y = -2$$ $$4x + 3y = -23$$ 2. **Choose a method:** We will use the substitution or elimination method. Here, elimination is convenient. 3. **Eliminate one variable:** Multiply the first equation by 3 and the second by 4 to align coefficients of $y$: $$3(-9x + 4y) = 3(-2) \Rightarrow -27x + 12y = -6$$ $$4(4x + 3y) = 4(-23) \Rightarrow 16x + 12y = -92$$ 4. **Subtract the second from the first to eliminate $y$:** $$(-27x + 12y) - (16x + 12y) = -6 - (-92)$$ $$-27x + 12y - 16x - 12y = -6 + 92$$ $$-43x = 86$$ 5. **Solve for $x$:** $$x = \frac{86}{-43} = -2$$ 6. **Substitute $x = -2$ into one original equation to find $y$:** Using the second equation: $$4(-2) + 3y = -23$$ $$-8 + 3y = -23$$ $$3y = -23 + 8 = -15$$ $$y = \frac{-15}{3} = -5$$ **Final answer:** $$x = -2, \quad y = -5$$