Chapter 7.5, Problem 11E

To determine

To find: the values of $x,y,\text{and}z$ .

Answer to Problem 11E

The solution is $x=-1,y=4,z=8$ .

Explanation of Solution

Given information:

A matrix equation is given as

  $\left[\begin{array}{ccc}4& 5& 4\\ 13& 15& 3y\\ 2& 2z-6& 0\end{array}\right]=\left[\begin{array}{ccc}4& 2x+7& 4\\ 13& 15& 12\\ 2& 3z-14& 0\end{array}\right]$

Concept used:

If two matrices of equal dimension are equal then the corresponding elements of the matrices are also equal.

Dimension of matrix with a rows and b columns is given by $a\times b$ .

Two matrices $A=\left[a_{ij}\right]$ and $B=\left[b_{ij}\right]$ are equal when they have same dimension and their corresponding entries are equal.

Calculation:

Consider the given matrix equation.

  $\left[\begin{array}{ccc}4& 5& 4\\ 13& 15& 3y\\ 2& 2z-6& 0\end{array}\right]=\left[\begin{array}{ccc}4& 2x+7& 4\\ 13& 15& 12\\ 2& 3z-14& 0\end{array}\right]$

Now, corresponding elements of the above two matrices are equal.

  $\begin{array}{l}2x+7=5\mathrm{......}(1)\\ 3y=12\mathrm{......}(2)\\ 2z-6=3z-14\mathrm{......}(3)\end{array}$

From equation (1),

  $\begin{array}{c}2x+7=5\\ 2x=5-7\\ 2x=-2\\ x=-1\end{array}$

From equation (2),

  $\begin{array}{l}3y=12\\ y=\frac{12}{3}\\ y=4\end{array}$

From equation (3),

  $\begin{array}{c}2z-6=3z-14\\ 2z-3z=-14+6\\ -z=-8\\ z=8\end{array}$

So, the unknown values can be obtained as follows:

  $\begin{array}{l}x=-1\\ y=4\\ z=8\end{array}$

Therefore, the values of $x,y\text{and}z$ is

  $x=-1,y=4,z=8$