Q43. - Algebra 2

Question

Solve for x and y in the matrix equation  $\left[\begin{array}{c}x\\ 7\end{array}\right]+\left[\begin{array}{c}3y\\ -x\end{array}\right]=\left[\begin{array}{c}16\\ 12\end{array}\right]$

Step-by-Step Solution

Answer

The values of x and y in the matrix equation $\left[\begin{array}{c}x\\ 7\end{array}\right]+\left[\begin{array}{c}3y\\ -x\end{array}\right]=\left[\begin{array}{c}16\\ 12\end{array}\right]$ is $\left(-5,7\right)$.

Step 1 - Define concept of matrix addition

If A and B are two matrices of order $m\times n$, then the addition of the two matrices will also yield a matrix of order $m\times n$wherein each element is the sum of the corresponding elements.

Step 2 - Add corresponding elements

In order to find x, first add corresponding elements of both the matrices 

$\left[\begin{array}{c}x\\ 7\end{array}\right]+\left[\begin{array}{c}3y\\ -x\end{array}\right].$

$\begin{array}{c}\left[\begin{array}{c}x\\ 7\end{array}\right]+\left[\begin{array}{c}3y\\ -x\end{array}\right]=\left[\begin{array}{c}16\\ 12\end{array}\right]\\ \left[\begin{array}{c}x+3y\\ 7-x\end{array}\right]=\left[\begin{array}{c}16\\ 12\end{array}\right]\end{array}$

Step 3 - Equate the matrix

In order to find the values of x and y, equate the corresponding elements since the two matrices are equal therefore, there corresponding elements will also be equal. 

 

$x+3y=16$   … (i)

$7-x=12$     …(ii)

Step 4 - Solve for x and y

Solve equation (ii) for x and get,

$\begin{array}{c}7-x-7=12-7\\ -x=5\\ -x\times \left(-1\right)=5\times \left(-1\right)\\ x=-5\end{array}$

Substitute -5 for x into equation (i) and solve for y.

$\begin{array}{c}\left(-5\right)+3y=16\\ \left(-5\right)+3y+5=16+5\\ \frac{3y}{3}=\frac{21}{3}\\ y=7\end{array}$

Therefore, the solution of the matrix is $\left(-5,7\right)$.