Finding x and y
Question:
Solve: \[ 2\begin{bmatrix}x & 5 \\ 7 & y-3\end{bmatrix} + \begin{bmatrix}3 & 4 \\ 1 & 2\end{bmatrix} = \begin{bmatrix}7 & 14 \\ 15 & 14\end{bmatrix} \]
Solve: \[ 2\begin{bmatrix}x & 5 \\ 7 & y-3\end{bmatrix} + \begin{bmatrix}3 & 4 \\ 1 & 2\end{bmatrix} = \begin{bmatrix}7 & 14 \\ 15 & 14\end{bmatrix} \]
Solution:
Step 1: Multiply
\[ 2\begin{bmatrix}x & 5 \\ 7 & y-3\end{bmatrix} = \begin{bmatrix}2x & 10 \\ 14 & 2y-6\end{bmatrix} \]Step 2: Add matrices
\[ = \begin{bmatrix} 2x+3 & 10+4 \\ 14+1 & (2y-6)+2 \end{bmatrix} = \begin{bmatrix} 2x+3 & 14 \\ 15 & 2y-4 \end{bmatrix} \]Step 3: Compare elements
\[ 2x + 3 = 7 \Rightarrow 2x = 4 \Rightarrow x = 2 \] \[ 2y – 4 = 14 \Rightarrow 2y = 18 \Rightarrow y = 9 \]Final Answer:
\[ x = 2,\quad y = 9 \]