Question

Find the value of $x+y$ from the following equation

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

In this question, We are given with a matrix equation

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