1. **State the problem:** Solve the system of linear equations: $$\begin{cases} a + b + c = 22.3 \\ 3a - b - 6c = -11.8 \\ 2a + b - c = 24.7 \end{cases}$$ 2. **Write the system in matrix form:** $$\left[\begin{array}{ccc|c} 1 & 1 & 1 & 22.3 \\ 3 & -1 & -6 & -11.8 \\ 2 & 1 & -1 & 24.7 \end{array}\right]$$ 3. **Use substitution or elimination to solve:** From the first equation: $$b = 22.3 - a - c$$ Substitute into the second and third equations: Second equation: $$3a - (22.3 - a - c) - 6c = -11.8$$ Simplify: $$3a - 22.3 + a + c - 6c = -11.8$$ $$4a - 22.3 - 5c = -11.8$$ $$4a - 5c = 22.3 - 11.8 = 10.5$$ Third equation: $$2a + (22.3 - a - c) - c = 24.7$$ Simplify: $$2a + 22.3 - a - c - c = 24.7$$ $$a + 22.3 - 2c = 24.7$$ $$a - 2c = 24.7 - 22.3 = 2.4$$ 4. **Solve the two equations:** $$\begin{cases} 4a - 5c = 10.5 \\ a - 2c = 2.4 \end{cases}$$ Multiply the second equation by 4: $$4a - 8c = 9.6$$ Subtract this from the first: $$ (4a - 5c) - (4a - 8c) = 10.5 - 9.6$$ $$4a - 5c - 4a + 8c = 0.9$$ $$3c = 0.9$$ $$c = 0.3$$ 5. **Find $a$ using $a - 2c = 2.4$:** $$a - 2(0.3) = 2.4$$ $$a - 0.6 = 2.4$$ $$a = 3$$ 6. **Find $b$ using $b = 22.3 - a - c$:** $$b = 22.3 - 3 - 0.3 = 19$$ **Final answer:** $$a = 3, \quad b = 19, \quad c = 0.3$$