Question

In the matrix equation below, what are the values of x and y?

Answer 1

Answer:

The values of x and y are 5 and 2 respectively.

Step-by-step explanation:

We have given a matrix equation.

$\frac{1}{2} \left[\begin{array}{ccc}4&8\\x+3&-4\\\end{array}\right] -3\left[\begin{array}{ccc}1&y+1\\-1&-2\end{array}\right] =\left[\begin{array}{ccc}-1&-5\\7&4\end{array}\right]$

We have to find the value of x and y.

Multiplying the scalars into the matrix, we have

$\left[\begin{array}{ccc}2&4\\\frac{x+3}{2}&-2\end{array}\right] -\left[\begin{array}{ccc}3&3y+3\\-3&-6\end{array}\right] =\left[\begin{array}{ccc}-1&-5\\7&4\end{array}\right]$

Adding the matrices of left sides, we have

$\left[\begin{array}{ccc}2-3&4-3y-3\\\frac{x+3}{2}+3&-2+6\end{array}\right]$ = $\left[\begin{array}{ccc}-1&-5\\7&4\end{array}\right]$

$\left[\begin{array}{ccc}-1&1-3y\\\frac{x+9}{2}&4\end{array}\right]$ =  $\left[\begin{array}{ccc}-1&-5\\7&4\end{array}\right]$

Two matrices are equal if their entries are equal.

1-3y = -5 ⇒ -3y = -5-1 ⇒ -3y  = -6

y = 2

$\frac{x+9}{2} =7$

x+9 = 14

x = 14-9

x = 5

Hence, the values of x and y are 5 and 2 respectively.

Answer 2

Answer:

$x=5,y=2$

Step-by-step explanation:

The given matrix equation is

$\frac{1}{2} \left[\begin{array}{cc}4&8\\x+3&-4\end{array}\right] -3\left[\begin{array}{cc}1&y+1\\-1&-2\end{array}\right]=\left[\begin{array}{cc}-1&-5\\7&4\end{array}\right]$

We multiply the scalars to get;

$\left[\begin{array}{cc}2&4\\ \frac{x+3}{2}&-2\end{array}\right] --\left[\begin{array}{cc}3&3y+3\\-3&-6\end{array}\right]=\left[\begin{array}{cc}-1&-5\\7&4\end{array}\right]$

We simplify the right hand side to get;

$\left[\begin{array}{cc}-1&1-3y\\ \frac{x+3}{2}+3&4\end{array}\right] =\left[\begin{array}{cc}-1&-5\\7&4\end{array}\right]$

The two matrices are equal. This means the corresponding entries are also equal.

$\Rightarrow 1-3y=-5$

$\Rightarrow -3y=-5-1$

$\Rightarrow -3y=-6$

$\Rightarrow y=2$

Also

$\frac{x+3}{2}+3=7$

$\Rightarrow \frac{x+3}{2}=7-3$

$\Rightarrow \frac{x+3}{2}=4$

$\Rightarrow x+3=8$

$\Rightarrow x=8-3$

$\Rightarrow x=5$