Question: Solve the following system. \[ \begin{cases} 3x - 3y + 4z = 10 \\ -5y + 6z = 4 \\ z = 4…

Solve the following system.

\[ \begin{cases} 3x - 3y + 4z = 10 \\ -5y + 6z = 4 \\ z = 4 \end{cases} \]

\( x = \boxed{\phantom{0}} \)

\( y = \boxed{\phantom{0}} \)

\( z = \boxed{\phantom{0}} \)

Solution

To solve the system of equations, we have: \[ \begin{cases} 3x - 3y + 4z = 10 & \\ -5y + 6z = 4 & \\ z = 4 & \end{cases} \] Start with the third equation to find \(z\): Substitute \(z = 4\) into the second equation: \[ -5y + 6(4) = 4 \] Simplify: \[ -5y + 24 = 4 \] Subtract 24 from both sides: \[ -5y = 4 - 24 \] \[ -5y = -20 \] Divide by -5: \[ y = 4 \] Now substitute \(y = 4\) and \(z = 4\) into the first equation: \[ 3x - 3(4) + 4(4) = 10 \] Simplify: \[ 3x - 12 + 16 = 10 \] Combine like terms: \[ 3x + 4 = 10 \] Subtract 4 from both sides: \[ 3x = 6 \] Divide by 3: \[ x = 2 \] The solution is: \(x = 2\), \(y = 4\), \(z = 4\)