Chapter 16.1, Problem 20WE

To determine

To find: the value of $x$ and $y$ in given equation.

Answer to Problem 20WE

The value of $x$ is $\frac{1-b}{a}$ , and that of $y$ is $-\frac{d}{c}$ .

Explanation of Solution

Given information:

Given symbol,

  $\left[\begin{array}{cc}ax+b& cy+d\end{array}\right]=\left[\begin{array}{cc}1& 0\end{array}\right]$

Let, $a,b,c$ and $d$ be nonzero real numbers.

Calculation:

If two matrices have the same dimensions and the elements in all corresponding positions are equal, then the matrices are equal.

Since the matrices are equal, equate the elements in corresponding positions.

  $ax+b=1,cy+d=0$

Solve for $x$ . Subtract $b$ from each of $ax+b=1$ .

  $\begin{array}{c}ax+b-b=1-b\\ ax=1-b\\ \frac{ax}{a}=\frac{1-b}{a}\\ x=\frac{1-b}{a}\end{array}$

Solve for $y$ . Subtract $d$ from each of $cy+d=0$ .

  $\begin{array}{c}cy+d=0\\ cy+d-d=0-d\\ cy=-d\\ \frac{cy}{c}=-\frac{d}{c}\\ y=-\frac{d}{c}\end{array}$

Therefore, the value of $x$ is $\frac{1-b}{a}$ , and that of $y$ is $-\frac{d}{c}$ .