1. **State the problem:** Solve the system of linear equations using Gauss elimination: $$\begin{cases}-2b + 10c = 4 \\ 3a + 12b - 6c = -9 \\ 9a + 34b - 7c = -19 \end{cases}$$ 2. **Rewrite the system in augmented matrix form:** $$\left[\begin{array}{ccc|c}0 & -2 & 10 & 4 \\ 3 & 12 & -6 & -9 \\ 9 & 34 & -7 & -19 \end{array}\right]$$ 3. **Swap rows to get a leading coefficient in the first row:** Swap row 1 and row 2: $$\left[\begin{array}{ccc|c}3 & 12 & -6 & -9 \\ 0 & -2 & 10 & 4 \\ 9 & 34 & -7 & -19 \end{array}\right]$$ 4. **Make the leading coefficient of row 1 equal to 1 by dividing row 1 by 3:** $$\left[\begin{array}{ccc|c}\cancel{3} \frac{3}{3} & \cancel{12} \frac{12}{3} & \cancel{-6} \frac{-6}{3} & \cancel{-9} \frac{-9}{3} \\ 0 & -2 & 10 & 4 \\ 9 & 34 & -7 & -19 \end{array}\right] = \left[\begin{array}{ccc|c}1 & 4 & -2 & -3 \\ 0 & -2 & 10 & 4 \\ 9 & 34 & -7 & -19 \end{array}\right]$$ 5. **Eliminate the first coefficient in row 3 by subtracting 9 times row 1 from row 3:** Row 3 = Row 3 - 9 * Row 1 $$\left[\begin{array}{ccc|c}1 & 4 & -2 & -3 \\ 0 & -2 & 10 & 4 \\ 9 - 9*1 & 34 - 9*4 & -7 - 9*(-2) & -19 - 9*(-3) \end{array}\right] = \left[\begin{array}{ccc|c}1 & 4 & -2 & -3 \\ 0 & -2 & 10 & 4 \\ 0 & -2 & 11 & 8 \end{array}\right]$$ 6. **Eliminate the second coefficient in row 3 by subtracting row 2 from row 3:** Row 3 = Row 3 - Row 2 $$\left[\begin{array}{ccc|c}1 & 4 & -2 & -3 \\ 0 & -2 & 10 & 4 \\ 0 & -2 - (-2) & 11 - 10 & 8 - 4 \end{array}\right] = \left[\begin{array}{ccc|c}1 & 4 & -2 & -3 \\ 0 & -2 & 10 & 4 \\ 0 & 0 & 1 & 4 \end{array}\right]$$ 7. **Back substitution:** From row 3: $$c = 4$$ From row 2: $$-2b + 10c = 4 \Rightarrow -2b + 10(4) = 4 \Rightarrow -2b + 40 = 4$$ $$-2b = 4 - 40 = -36 \Rightarrow b = \frac{-36}{-2} = 18$$ From row 1: $$a + 4b - 2c = -3 \Rightarrow a + 4(18) - 2(4) = -3$$ $$a + 72 - 8 = -3 \Rightarrow a + 64 = -3 \Rightarrow a = -3 - 64 = -67$$ **Final solution:** $$a = -67, \quad b = 18, \quad c = 4$$