# How do you solve the system of equations algebraically 3x-5y+z=9, x-3y-2z=-8, 5x-6y+3z=15?

Jul 8, 2017

$\left(x , y , z\right) \to \left(- 6 , - 4 , 7\right)$

#### Explanation:

$\text{using the process of elimination}$

$\textcolor{b l u e}{\text{ eliminating z}}$

$3 x - 5 y + z = 9 \to \left(1\right)$

$x - 3 y - 2 z = - 8 \to \left(2\right)$

$5 x - 6 y + 3 z = 15 \to \left(3\right)$

$\left(2\right) + 2 \times \left(1\right) \leftarrow \text{ row operation}$

$7 x - 13 y = 10 \to \left(4\right)$

$\left(3\right) - 3 \times \left(1\right) \leftarrow \text{ row operation}$

$- 4 x + 9 y = - 12 \to \left(5\right)$

$\textcolor{b l u e}{\text{ eliminating x}}$

$4 \times \left(4\right) + 7 \times \left(5\right) \leftarrow \text{ row operation}$

$11 y = - 44 \to \textcolor{red}{y = - 4}$

$\text{substituting in } \left(4\right)$

$7 x + 52 = 10 \to \textcolor{red}{x = - 6}$

$\text{substituting in } \left(1\right)$

$- 18 + 20 + z = 9 \to \textcolor{red}{z = 7}$

$\textcolor{b l u e}{\text{As a check"" in }} \left(2\right)$

$- 6 + 12 - 14 = - 8 \rightarrow \text{ True}$

$\Rightarrow \left(- 6 , - 4 , 7\right) \text{ is the solution}$