Chapter 11, Problem 14CT

To determine

To find: The solution of the system of equations.

Answer to Problem 14CT

The solution of the system of equations is inconsistent.

Explanation of Solution

Given information:

The system of equations is

  $\{\begin{array}{l}x+\frac{1}{4}y=7\\ 8x+2y=56\end{array}$

Calculation:

Write the system of equations into the matrices.

  $A=\left[\begin{array}{cc}1& \frac{1}{4}\\ 8& 2\end{array}\right]$

  $X=\left[\begin{array}{c}x\\ y\end{array}\right]$

  $B=\left[\begin{array}{c}7\\ 56\end{array}\right]$

Write the original equation in the compact form.

  $AX=B$

Calculate the matrix $X$ in the above equation.

  $\begin{array}{c}{A}^{-1}\left(AX\right)=A^{-1}B\\ \left(A^{-1}A\right)X=A^{-1}B\\ I_{2}X=A^{-1}B\\ X=A^{-1}B\end{array}$

The inverse of the matrix $A$ cannot be determined since the matrix of the coefficient of the variables in the given equations is singular. Hence the system of equations are inconsistent.

Therefore, the solution of the system of equations is inconsistent.