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.)

\[ \begin{cases} 2x + y - z = -8 \\ 3x + 2y + 3z = 21 \\ 7x + 4y + z = 5 \end{cases} \]

\((x, y, z) = (\boxed{\phantom{0}}, \boxed{\phantom{0}}, \boxed{\phantom{0}})\)

Solution

Step 1 \[ 2x + y - z = -8 & \] Step 2 \[ 3x + 2y + 3z = 21 & \] Step 3 \[ 7x + 4y + z = 5 & \] Step 4 Solve the first equation for \( z \): \[ z = 2x + y + 8 & \] Step 5 Substitute \( z = 2x + y + 8 \) into the second equation: \[ 3x + 2y + 3(2x + y + 8) = 21 & \] Simplify: \[ 9x + 5y = -3 & \] Step 6 Substitute \( z = 2x + y + 8 \) into the third equation: \[ 7x + 4y + (2x + y + 8) = 5 & \] Simplify: \[ 9x + 5y = -3 & \] Step 7 Since both the second and third equations simplify to \( 9x + 5y = -3 \), the system is dependent. Step 8 Let \( x = c \). Then, \[ y = \frac{-9c - 3}{5} & \] Step 9 Substitute \( x = c \) and \( y = \frac{-9c - 3}{5} \) into \( z = 2x + y + 8 \): \[ z = \frac{c + 37}{5} & \] Solution: \[ (x, y, z) = \left( c, \frac{-9c - 3}{5}, \frac{c + 37}{5} \right) \]