Given the system of equations: \( \left\{\begin{array}{l}2 x-3 y-9 z=30 \\ x+3 z=-12 \\ -3 x+y-4 z=18\end{array}\right. \) (a) Determine the type of system: O dependentO inconsistent (b) If your answer is dependent in (a), find the complete solution. Write \( x, y \) as functions of \( t \), where \( z=t \). If your answer is inconsistent, write DNE for all three variables. \( x=\square \) \( y=\square \)

Solve the system of equations \( 2x-3y-9z=30;x+3z=-12;-3x+y-4z=18 \). Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}2x-3y-9z=30\\x+3z=-12\\-3x+y-4z=18\end{array}\right.\) - step1: Solve the equation: \(\left\{ \begin{array}{l}2x-3y-9z=30\\x=-12-3z\\-3x+y-4z=18\end{array}\right.\) - step2: Substitute the value of \(x:\) \(\left\{ \begin{array}{l}2\left(-12-3z\right)-3y-9z=30\\-3\left(-12-3z\right)+y-4z=18\end{array}\right.\) - step3: Simplify: \(\left\{ \begin{array}{l}-24-15z-3y=30\\36+5z+y=18\end{array}\right.\) - step4: Solve the equation: \(\left\{ \begin{array}{l}-24-15z-3y=30\\y=-18-5z\end{array}\right.\) - step5: Substitute the value of \(y:\) \(-24-15z-3\left(-18-5z\right)=30\) - step6: Simplify: \(30=30\) - step7: Calculate: \(\left(x,y,z\right) = \left(-12-3z,-18-5z,z\right),z \in \mathbb{R}\) - step8: Alternative Form: \(\textrm{Infinitely many solutions}\) The system of equations is dependent, meaning it has infinitely many solutions. To find the complete solution, we can express \( x \) and \( y \) as functions of \( t \), where \( z=t \). Let's denote the solutions as \( x = x(t) \) and \( y = y(t) \). Since the system is dependent, we can express one variable in terms of the other variables. Let's solve for \( x \) in terms of \( t \) first. From the second equation \( x + 3z = -12 \), we can express \( x \) in terms of \( z \): \[ x = -3z - 12 \] Substitute this expression for \( x \) into the first equation \( 2x - 3y - 9z = 30 \): \[ 2(-3z - 12) - 3y - 9z = 30 \] \[ -6z - 24 - 3y - 9z = 30 \] \[ -15z - 3y = 54 \] \[ -5z - y = 18 \] \[ y = -5z - 18 \] Therefore, the complete solution is: \[ x = -3z - 12 \] \[ y = -5z - 18 \] So, \( x = -3z - 12 \) and \( y = -5z - 18 \).