Chapter 4.5, Problem 52PPS

To determine

To find: The solution of equations using Cramer’s rule.

Answer to Problem 52PPS

The solution of the system is $\left(\frac{2072}{271},-6,-3\right)$ .

Explanation of Solution

Given,

The set of equation is given as,

  $\begin{array}{l}4x-5y=-2\\ 7x+3z=-47\\ 8y-5z=-63\end{array}$

Calculation:

Calculate the determinant of x .

  $\begin{array}{c}\left|x\right|=\left|\begin{array}{ccc}-2& 5& 0\\ -47& 0& 3\\ -63& 8& -5\end{array}\right|\\ =-2\left(0-24\right)-5\left(235+189\right)+0\\ =-2072\end{array}$

Calculate the determinant of y .

  $\begin{array}{c}\left|y\right|=\left|\begin{array}{ccc}4& -2& 0\\ 7& -47& 3\\ 0& -63& -5\end{array}\right|\\ =4\left(235+189\right)+2\left(-35-0\right)+0\\ =1626\end{array}$

Calculate the determinant of z .

  $\begin{array}{c}\left|z\right|=\left|\begin{array}{ccc}4& -5& -2\\ 7& 0& -47\\ 0& 8& -63\end{array}\right|\\ =4\left(0+376\right)+5\left(-441-0\right)-2\left(56-0\right)\\ =-813\end{array}$

Calculate the determinant C .

  $\begin{array}{c}\left|C\right|=\left|\begin{array}{ccc}4& -5& 0\\ 7& 0& 3\\ 0& 8& -5\end{array}\right|\\ =4\left(0-24\right)+5\left(-35-0\right)+0\\ =-271\end{array}$

Calculate the value of x, yandz.

  $\begin{array}{c}x=\frac{-2072}{-271}\\ y=\frac{1626}{-271}\\ =-6\\ z=\frac{813}{-271}\\ =-3\end{array}$

Thus, the solution of the system is $\left(\frac{2072}{271},-6,-3\right)$ .