1. **State the problem:** Solve the system of linear equations: $$\begin{cases} x + 6y = -9 \\ -x + 3y = 9 \end{cases}$$ 2. **Add the two equations to eliminate $x$: ** $$ (x + 6y) + (-x + 3y) = -9 + 9 $$ $$ \cancel{x} + 6y - \cancel{x} + 3y = 0 $$ $$ 9y = 0 $$ 3. **Solve for $y$: ** $$ y = \frac{0}{9} = 0 $$ 4. **Substitute $y=0$ into the first equation to find $x$: ** $$ x + 6(0) = -9 $$ $$ x = -9 $$ 5. **Final solution:** $$ (x, y) = (-9, 0) $$ This means the two lines intersect at the point $(-9, 0)$.