Finding x and y
Question:
Solve: \[ 2\begin{bmatrix}3 & 4 \\ 5 & x\end{bmatrix} + \begin{bmatrix}1 & y \\ 0 & 1\end{bmatrix} = \begin{bmatrix}7 & 0 \\ 10 & 5\end{bmatrix} \]
Solve: \[ 2\begin{bmatrix}3 & 4 \\ 5 & x\end{bmatrix} + \begin{bmatrix}1 & y \\ 0 & 1\end{bmatrix} = \begin{bmatrix}7 & 0 \\ 10 & 5\end{bmatrix} \]
Solution:
Step 1: Multiply the matrix
\[ 2\begin{bmatrix}3 & 4 \\ 5 & x\end{bmatrix} = \begin{bmatrix}6 & 8 \\ 10 & 2x\end{bmatrix} \]Step 2: Add matrices
\[ = \begin{bmatrix} 6+1 & 8+y \\ 10+0 & 2x+1 \end{bmatrix} = \begin{bmatrix} 7 & 8+y \\ 10 & 2x+1 \end{bmatrix} \]Step 3: Compare corresponding elements
\[ 8 + y = 0 \Rightarrow y = -8 \] \[ 2x + 1 = 5 \Rightarrow 2x = 4 \Rightarrow x = 2 \]Final Answer:
\[ x = 2, \quad y = -8 \]