Chapter 7, Problem 21RE

Solution

To calculate: The solution of the system is exists or not, if does, then find the solution.

  $\begin{array}{c}x+2y=1\\ 4y-4=-2x\end{array}$

The solution does not exist.

Given Information:

The system of equation is defined as,

  $\begin{array}{c}x+2y=1\\ 4y-4=-2x\end{array}$

Formula Used:

The inverse of 2 cross 2 matrix is defined as,

  ${\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]}^{-1}=\frac{1}{ad-bc}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$

Calculation:

Consider the given equation,

  $\begin{array}{c}x+2y=1\\ 4y-4=-2x\end{array}$

Simplify the given equation.

  $\begin{array}{c}x+2y=1\\ 2x+4y=4\end{array}$

Rewrite the given system as matrix form.

  $\begin{array}{c}AX=B\\ \left[\begin{array}{cc}1& 2\\ 2& 4\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}1\\ 4\end{array}\right]\end{array}$

Find the inverse of the matrix $A$ .

  $\begin{array}{c}{A}^{-1}={\left[\begin{array}{cc}1& 2\\ 2& 4\end{array}\right]}^{-1}\\ =\frac{1}{4-4}\left[\begin{array}{cc}1& 2\\ 2& 4\end{array}\right]\\ =\text{undefined}\end{array}$

As the inverse of the matrix is undefined this implies the solution does not exist.

Hence, the solution does not exists.