Chapter 12.1, Problem 32PPE

To determine

To find: the value of $x$ and $y$ so that the given matrix equation becomes true.

Answer to Problem 32PPE

  $x=4,y=-2$ .

Explanation of Solution

Given:

The given matrix equation is $\left[\begin{array}{l}x-1\\ 43x\end{array}\right]+\left[\begin{array}{l}-3y5\\ -62y\end{array}\right]=\left[\begin{array}{l}74\\ -210\end{array}\right]$ .

Concept used:

The sum or differences of the any matrix is possible, when the number of elements of both matrices are same. i.e. number of columns are same and number of rows are same.

Calculation:

As the given matrix equation is $\left[\begin{array}{l}x-1\\ 43x\end{array}\right]+\left[\begin{array}{l}-3y5\\ -62y\end{array}\right]=\left[\begin{array}{l}74\\ -210\end{array}\right]$ .

Here number of elements in both matrices are same.

Therefore, add the matrix. Now,

  $\begin{array}{l}\left[\begin{array}{l}x-1\\ 43x\end{array}\right]+\left[\begin{array}{l}-3y5\\ -62y\end{array}\right]=\left[\begin{array}{l}74\\ -210\end{array}\right]\\ \Rightarrow \left[\begin{array}{l}x+\left(-3y\right)-1+5\\ 4+\left(-6\right)3x+2y\end{array}\right]=\left[\begin{array}{l}74\\ -210\end{array}\right]\left\{\text{add the matrix}\right\}\\ \Rightarrow \left[\begin{array}{l}x-3y4\\ -23x+2y\end{array}\right]=\left[\begin{array}{l}74\\ -210\end{array}\right]\end{array}$

On comparing the elements of the LHS and RHS of the given matrix, it is found that

  $\begin{array}{l}x-3y=7\mathrm{......}(1)\\ 3x+2y=10\mathrm{......}(2)\end{array}$

Now, compute the value of $x$ from eq (1),

  $\begin{array}{l}x-3y=7\\ \Rightarrow x=7+3y\end{array}$

Now, substitutes the value in eq (2), then

  $\begin{array}{l}3\left(7+3y\right)+2y=10\text{ {sustitutes the value of }x\text{}}\\ 21+9y+2y=10\text{ {open the parenthesis}}\\ \therefore 11y=10-21\text{ {subtract 21 from both sides}}\\ 11y=-11\\ \therefore y=\frac{-11}{11}\text{ {divide }11\text{ from both sides}}\\ \Rightarrow y=-1\end{array}$

Now, substitutes the value of y in eq {1}

  $\begin{array}{l}x=7+3\left(-1\right)\{y=-1\}\\ =7-3=-4\end{array}$

Hence, the value of $x$ and $y$ so that the equation becomes true is $x=4,y=-2$ .