Finding x, y, z, t
Question:
Solve: \[ 3\begin{bmatrix}x & y \\ z & t\end{bmatrix} = \begin{bmatrix}x & 6 \\ -1 & 2t\end{bmatrix} + \begin{bmatrix}4 & x+y \\ z+t & 3\end{bmatrix} \]
Solve: \[ 3\begin{bmatrix}x & y \\ z & t\end{bmatrix} = \begin{bmatrix}x & 6 \\ -1 & 2t\end{bmatrix} + \begin{bmatrix}4 & x+y \\ z+t & 3\end{bmatrix} \]
Solution:
Step 1: LHS
\[ = \begin{bmatrix} 3x & 3y \\ 3z & 3t \end{bmatrix} \]Step 2: RHS (Add matrices)
\[ = \begin{bmatrix} x+4 & 6 + x + y \\ -1 + z + t & 2t + 3 \end{bmatrix} \]Step 3: Compare elements
\[ 3x = x + 4 \Rightarrow 2x = 4 \Rightarrow x = 2 \] \[ 3y = 6 + x + y \Rightarrow 3y = 6 + 2 + y \Rightarrow 2y = 8 \Rightarrow y = 4 \] \[ 3t = 2t + 3 \Rightarrow t = 3 \] \[ 3z = -1 + z + t \Rightarrow 3z = -1 + z + 3 \Rightarrow 2z = 2 \Rightarrow z = 1 \]Final Answer:
\[ x = 2,\quad y = 4,\quad z = 1,\quad t = 3 \]