1. **State the problem:** Solve the system of linear equations: $$13x - 6y = -2$$ $$8x + 3y = -22$$ 2. **Choose a method:** We will use the elimination method to eliminate one variable. 3. **Make coefficients of $y$ opposites:** Multiply the first equation by 3 and the second equation by 6: $$3(13x - 6y) = 3(-2) \Rightarrow 39x - 18y = -6$$ $$6(8x + 3y) = 6(-22) \Rightarrow 48x + 18y = -132$$ 4. **Add the two equations to eliminate $y$:** $$39x - 18y + 48x + 18y = -6 - 132$$ $$ (39x + 48x) + (-18y + 18y) = -138$$ $$87x + \cancel{-18y + 18y} = -138$$ $$87x = -138$$ 5. **Solve for $x$:** $$x = \frac{-138}{87} = \frac{\cancel{-138}^{-46} \times 3}{\cancel{87}^{29} \times 3} = -\frac{46}{29}$$ 6. **Substitute $x$ back into one of the original equations to find $y$:** Use the second equation: $$8x + 3y = -22$$ $$8\left(-\frac{46}{29}\right) + 3y = -22$$ $$-\frac{368}{29} + 3y = -22$$ 7. **Isolate $y$:** $$3y = -22 + \frac{368}{29}$$ Convert $-22$ to a fraction with denominator 29: $$-22 = -\frac{638}{29}$$ $$3y = -\frac{638}{29} + \frac{368}{29} = -\frac{270}{29}$$ 8. **Solve for $y$:** $$y = \frac{-\frac{270}{29}}{3} = -\frac{270}{29} \times \frac{1}{3} = -\frac{270}{87} = -\frac{\cancel{270}^{90} \times 3}{\cancel{87}^{29} \times 3} = -\frac{90}{29}$$ **Final answer:** $$x = -\frac{46}{29}, \quad y = -\frac{90}{29}$$