Question

Find the value of \[x+y\] from the following equation
\[2\left[ \begin{matrix}
   x & 5 \\
   7 & y-3 \\
\end{matrix} \right]+\left[ \begin{matrix}
   3 & -4 \\
   1 & 2 \\
\end{matrix} \right]=\left[ \begin{matrix}
   7 & 6 \\
   15 & 14 \\
\end{matrix} \right]\]

Answer 1

Hint: In this question, We are given with a matrix equation
\[2\left[ \begin{matrix}
   x & 5 \\
   7 & y-3 \\
\end{matrix} \right]+\left[ \begin{matrix}
   3 & -4 \\
   1 & 2 \\
\end{matrix} \right]=\left[ \begin{matrix}
   7 & 6 \\
   15 & 14 \\
\end{matrix} \right]\]. In order to find the value of \[x+y\], we have to first calculate the value of \[x\] and \[y\]. In order to find \[x\] and \[y\], we have to first multiply value of \[2\left[ \begin{matrix}
   x & 5 \\
   7 & y-3 \\
\end{matrix} \right]\]
 by multiplying each element of the matrix with 2. Then we have to add the resultant matrix to
\[\left[ \begin{matrix}
   3 & -4 \\
   1 & 2 \\
\end{matrix} \right]\] by using component wise addition. Finally we have to equate each of term of the resultant matrix with the corresponding terms of \[\left[ \begin{matrix}
   7 & 6 \\
   15 & 14 \\
\end{matrix} \right]\] and form equations in \[x\] and \[y\] and solve those equations to get the desired value of \[x\]and \[y\].Then we will add both the value to get
\[x+y\].

Complete step-by-step answer:
We are given with a matrix equation \[2\left[ \begin{matrix}
   x & 5 \\
   7 & y-3 \\
\end{matrix} \right]+\left[ \begin{matrix}
   3 & -4 \\
   1 & 2 \\
\end{matrix} \right]=\left[ \begin{matrix}
   7 & 6 \\
   15 & 14 \\
\end{matrix} \right].........(1)\].
Now in order to find the value of \[x+y\], we will first have to first calculate the value of \[x\]and \[y\].
For that we will first calculate the matrix \[2\left[ \begin{matrix}
   x & 5 \\
   7 & y-3 \\
\end{matrix} \right]\] by multiplying each element of the matrix
\[\left[ \begin{matrix}
   x & 5 \\
   7 & y-3 \\
\end{matrix} \right]\] with 2.
Then we will get
\[\begin{align}
  & 2\left[ \begin{matrix}
   x & 5 \\
   7 & y-3 \\
\end{matrix} \right]=\left[ \begin{matrix}
   2x & 2\times 5 \\
   2\times 7 & 2\left( y-3 \right) \\
\end{matrix} \right] \\
 & =\left[ \begin{matrix}
   2x & 10 \\
   14 & 2y-6 \\
\end{matrix} \right]
\end{align}\]
Now on adding the matrix value of \[2\left[ \begin{matrix}
   x & 5 \\
   7 & y-3 \\
\end{matrix} \right]\] to the matrix \[\left[ \begin{matrix}
   3 & -4 \\
   1 & 2 \\
\end{matrix} \right]\] by component wise addition of the elements of the matrices, we will get

\[\begin{align}
  & 2\left[ \begin{matrix}
   x & 5 \\
   7 & y-3 \\
\end{matrix} \right]+\left[ \begin{matrix}
   3 & -4 \\
   1 & 2 \\
\end{matrix} \right]=\left[ \begin{matrix}
   2x & 10 \\
   14 & 2y-6 \\
\end{matrix} \right]+\left[ \begin{matrix}
   3 & -4 \\
   1 & 2 \\
\end{matrix} \right] \\
 & =\left[ \begin{matrix}
   2x+3 & 10-4 \\
   14+1 & 2y-6+2 \\
\end{matrix} \right] \\
 & =\left[ \begin{matrix}
   2x+3 & 6 \\
   15 & 2y-4 \\
\end{matrix} \right]
\end{align}\]
Now we will substitute that value of \[2\left[ \begin{matrix}
   x & 5 \\
   7 & y-3 \\
\end{matrix} \right]+\left[ \begin{matrix}
   3 & -4 \\
   1 & 2 \\
\end{matrix} \right]\]in the matrix equation (1), then we get
\[\left[ \begin{matrix}
   2x+3 & 6 \\
   15 & 2y-4 \\
\end{matrix} \right]=\left[ \begin{matrix}
   7 & 6 \\
   15 & 14 \\
\end{matrix} \right]\]
Equation each elements of the two matrices in the right hand side and the left hand side of the above matrix equation, we get
\[2x+3=7.............(2)\] and
\[2y-4=14.............(3)\]
On solving equation (2) to find the value of \[x\], we will get

\[\begin{align}
  & 2x=7-3 \\
 & \Rightarrow 2x=4 \\
 & \Rightarrow x=\dfrac{4}{2} \\
 & \Rightarrow x=2 \\
\end{align}\]
Now On solving equation (3) to find the value of \[y\], we will get
\[\begin{align}
  & 2y-4=14 \\
 & \Rightarrow 2y=14+4 \\
 & \Rightarrow 2y=18 \\
 & \Rightarrow y=\dfrac{18}{2} \\
 & \Rightarrow y=9 \\
\end{align}\]
Thus we have \[x=2\] and \[y=9\].
On adding the values \[x=2\] and \[y=9\] to get \[x+y\], we get
\[\begin{align}
  & x+y=2+9 \\
 & =11
\end{align}\]
Thus we get that the value of \[x+y\] is equal to 11.

Note: In this problem, while calculating the product \[2\left[ \begin{matrix}
   x & 5 \\
   7 & y-3 \\
\end{matrix} \right]\], 2 will be multiplied by each and every element of the matrix and not just the element in the first row and first column.