1. **State the problem:** We have the homogeneous system of equations: $$x + 3y = 0$$ $$2x + 6y = 0$$ We want to prove using the augmented matrix that the system has nontrivial solutions and find two different solutions. 2. **Write the augmented matrix:** $$\left[\begin{array}{cc|c} 1 & 3 & 0 \\ 2 & 6 & 0 \end{array}\right]$$ 3. **Perform row operations to reduce the matrix:** Replace row 2 by row 2 minus 2 times row 1: $$\left[\begin{array}{cc|c} 1 & 3 & 0 \\ 2 & 6 & 0 \end{array}\right] \xrightarrow{R_2 \to R_2 - 2R_1} \left[\begin{array}{cc|c} 1 & 3 & 0 \\ \cancel{2} & \cancel{6} & 0 \end{array}\right] = \left[\begin{array}{cc|c} 1 & 3 & 0 \\ 0 & 0 & 0 \end{array}\right]$$ 4. **Interpret the reduced matrix:** The second row is all zeros, indicating the system has infinitely many solutions. 5. **Express variables:** From the first row: $$x + 3y = 0 \implies x = -3y$$ 6. **Find two different solutions:** Choose arbitrary values for $y$: - For $y=1$, $x = -3(1) = -3$, so solution is $(-3,1)$. - For $y=2$, $x = -3(2) = -6$, so solution is $(-6,2)$. 7. **Conclusion:** The system has nontrivial solutions because the determinant of the coefficient matrix is zero and the system is dependent. **Final answer:** Two different solutions are $(-3,1)$ and $(-6,2)$.