1. **State the problem:** Solve the system of linear equations: $$\begin{cases} x + y - 9z + w = -8 \\ 3y - 5z - w = 2 \\ 6z - 7w = 7 \\ 3w = 6 \end{cases}$$ 2. **Start with the simplest equation:** From the fourth equation, solve for $w$: $$3w = 6$$ Divide both sides by 3: $$\cancel{3}w = \frac{6}{\cancel{3}} \Rightarrow w = 2$$ 3. **Substitute $w=2$ into the third equation:** $$6z - 7(2) = 7$$ Simplify: $$6z - 14 = 7$$ Add 14 to both sides: $$6z = 21$$ Divide both sides by 6: $$\cancel{6}z = \frac{21}{\cancel{6}} \Rightarrow z = \frac{21}{6} = \frac{7}{2} = 3.5$$ 4. **Substitute $w=2$ and $z=3.5$ into the second equation:** $$3y - 5(3.5) - 2 = 2$$ Calculate: $$3y - 17.5 - 2 = 2$$ $$3y - 19.5 = 2$$ Add 19.5 to both sides: $$3y = 21.5$$ Divide both sides by 3: $$\cancel{3}y = \frac{21.5}{\cancel{3}} \Rightarrow y = \frac{21.5}{3} = 7.1667$$ 5. **Substitute $y=7.1667$, $z=3.5$, and $w=2$ into the first equation:** $$x + 7.1667 - 9(3.5) + 2 = -8$$ Calculate: $$x + 7.1667 - 31.5 + 2 = -8$$ Simplify: $$x - 22.3333 = -8$$ Add 22.3333 to both sides: $$x = 14.3333$$ **Final solution:** $$x = 14.3333, \quad y = 7.1667, \quad z = 3.5, \quad w = 2$$