Question: Solve the system of equations. (If the system is dependent, enter a general solution in terms of…

Solve the system of equations. (If the system is dependent, enter a general solution in terms of \( c \). If there is no solution, enter NO SOLUTION.)

\[ \begin{cases} x - 3y + 2z = -7 \\ 3x + y + 4z = 2 \\ 5x - 5y + 8z = -12 \end{cases} \]

\((x, y, z) = (\boxed{\phantom{xxxxx}}, \boxed{\phantom{xxxxx}}, \boxed{\phantom{xxxxx}})\)

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Solution

To solve the system of equations, follow these steps: \[ \begin{cases} x - 3y + 2z = -7 & \\ 3x + y + 4z = 2 & \\ 5x - 5y + 8z = -12 & \end{cases} \] Multiply the first equation by 3: \[ 3x - 9y + 6z = -21 \] Subtract the second equation from this result: \[ -10y + 2z = -23 \] Solve for \( z \) in terms of \( y \): \[ z = 5y - \frac{23}{2} \] Substitute \( z \) into the first equation to solve for \( x \): \[ x = 16 - 7y \] Let \( y = c \), where \( c \) is a parameter. Then: \[ x = 16 - 7c \] \[ z = 5c - \frac{23}{2} \] The general solution is: \[ (x, y, z) = \left(16 - 7c, \, c, \, 5c - \frac{23}{2}\right) \]