Solve each equation

$\left[\begin{array}{c}\text{x}+3\text{y}\\ 2\text{x}-\text{y}\end{array}\right]=\left[\begin{array}{c}-22\\ 19\end{array}\right]$

A rectangular array of numbers arranged in rows and columns is called a matrix

Using the definition of equal matices is used to solve the equation.

If the dimension of the matrices are same and corresponding elements are equal then the matrices are equal.

Here the matrices are:

$\left[\begin{array}{c}x+3y\\ 2x-y\end{array}\right]=\left[\begin{array}{c}-22\\ 19\end{array}\right]$

Since, the matrices are equal then corresponding elements are also equal.

Then, the two linear equations formed are:

$x+3y=-22$ (1)

$2x-y=19$ (2)

Solve the equations using substitution.

Solve equation 2 as:

$\begin{array}{c}2x-y=19\\ 2x-y-19=19-19\\ 2x-19-y=0\\ 2x-19-y+y=0+y\\ y=2x-19\end{array}$

Substitute y with $y=2x-19$ in equation (1):

$\begin{array}{c}x+3y=-22\\ x+3(2x-19)=-22\\ x+6x-57=-22\\ 7x-57+57=-22+57\\ 7x=35\\ x=5\end{array}$

Now substitute value of in equation (1),

$\begin{array}{c}x+3y=-22\\ 5+3y=-22\\ 5+3y-5=-22-5\\ 3y=-27\\ y=-9\end{array}$

Therefore, $x=5$ and $y=-9$. The solution is $(5,-9)$