Chapter 10.4, Problem 37E

To determine

To calculate:

The value of $x$ and $y$

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

Answer to Problem 37E

The values are $x=1,y=-2$

Explanation of Solution

Given information:

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

Calculation:

Know that:

Equality of Matrices: Two matrices $A$ and $B$ are said to be equal only if $A$ and $B$ have the same order and their corresponding elements are equal.

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

Multiply the matrix on the left side by the scalar $2$ .

  $\begin{array}{l}2\left[\begin{array}{cc}x& y\\ x+y& x-y\end{array}\right]\\ =\left[\begin{array}{cc}2x& 2y\\ 2\left(x+y\right)& 2\left(x-y\right)\end{array}\right]=\left[\begin{array}{cc}2& -4\\ -2& 6\end{array}\right]\end{array}$

The matrix on the left side is equal to the matrix on the right side, this means that the entries in the same positions are equal.

  $\begin{array}{l}2x=2\\ 2y=-4\\ 2\left(x+y\right)=-2\\ 3\left(x+y\right)=6\end{array}$

Now Simplify,

  $\begin{array}{l}x=1\\ y=-2\\ x+y=-1\\ x-y=3\end{array}$

So now first $x=1$ and $y=-2$ into the third equation to see if both values on the left and the right side of the equality are the same.

  $1+\left(-2\right)=-1$ this is true.

Now substitute $x=1$ and $y=-2$ into the fourth equation to see if both values on the left and the right side of the equality are the same.

  $1-\left(-2\right)=3$ this is true.

So, the values are $x=1,y=-2$ .