1. Stating the problem: Solve the system of linear equations: $$-5x + 9y - 7y = 0$$ $$9x + 9y + 9z = 0$$ $$7x - y + 3z = -36$$ 2. Simplify the first equation by combining like terms: $$-5x + (9y - 7y) = 0 \Rightarrow -5x + 2y = 0$$ 3. Express $y$ in terms of $x$ from the first equation: $$2y = 5x \Rightarrow y = \frac{5}{2}x$$ 4. Substitute $y = \frac{5}{2}x$ into the second equation: $$9x + 9\left(\frac{5}{2}x\right) + 9z = 0 \Rightarrow 9x + \frac{45}{2}x + 9z = 0$$ 5. Combine $x$ terms: $$9x + \frac{45}{2}x = \frac{18}{2}x + \frac{45}{2}x = \frac{63}{2}x$$ 6. So the second equation reduces to: $$\frac{63}{2}x + 9z = 0 \Rightarrow 9z = -\frac{63}{2}x \Rightarrow z = -\frac{63}{18}x = -\frac{7}{2}x$$ 7. Substitute $y = \frac{5}{2}x$ and $z = -\frac{7}{2}x$ into the third equation: $$7x - \left(\frac{5}{2}x\right) + 3\left(-\frac{7}{2}x\right) = -36$$ 8. Calculate step by step: $$7x - \frac{5}{2}x - \frac{21}{2}x = -36$$ Combine terms: $$7x = \frac{14}{2}x$$ $$\frac{14}{2}x - \frac{5}{2}x - \frac{21}{2}x = -36$$ $$\frac{14 - 5 - 21}{2}x = -36$$ $$\frac{-12}{2}x = -36$$ $$-6x = -36$$ 9. Solve for $x$: $$x = \frac{-36}{-6} = 6$$ 10. Find $y$ using $y = \frac{5}{2}x$: $$y = \frac{5}{2} \times 6 = 15$$ 11. Find $z$ using $z = -\frac{7}{2}x$: $$z = -\frac{7}{2} \times 6 = -21$$ Final solution: $$x = 6,\quad y = 15,\quad z = -21$$