Chapter 4.5, Problem 51PPS

To determine

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

Answer to Problem 51PPS

The solution of the system is $\left(4,8,-5\right)$ .

Explanation of Solution

Given,

The set of equation is given as,

  $\begin{array}{l}3a-5b-9c=17\\ 4a-3c=31\\ -5a-4b-2c=-42\end{array}$

Calculation:

Calculate the determinant of x .

  $\begin{array}{c}\left|x\right|=\left|\begin{array}{ccc}17& -5& -9\\ 31& 0& -3\\ -42& -4& -2\end{array}\right|\\ =17\left(0-12\right)+5\left(-62-126\right)-9\left(-124-0\right)\\ =-28\end{array}$

Calculate the determinant of y .

  $\begin{array}{c}\left|y\right|=\left|\begin{array}{ccc}3& 17& -9\\ 4& 31& -3\\ -5& -42& -2\end{array}\right|\\ =3\left(-62-126\right)-17\left(-8-15\right)-9\left(-168+155\right)\\ =-56\end{array}$

Calculate the determinant of z .

  $\begin{array}{c}\left|z\right|=\left|\begin{array}{ccc}3& -5& 17\\ 4& 0& 31\\ -5& -4& -42\end{array}\right|\\ =3\left(0+124\right)+5\left(-168+155\right)+17\left(-16+0\right)\\ =35\end{array}$

Calculate the determinant C .

  $\begin{array}{c}\left|C\right|=\left|\begin{array}{ccc}3& -5& -9\\ 4& 0& -3\\ -5& -4& -2\end{array}\right|\\ =3\left(0-12\right)+5\left(-8-15\right)-9\left(-16-0\right)\\ =-7\end{array}$

Calculate the value of x, yandz.

  $\begin{array}{c}x=\frac{-28}{-7}\\ =4\\ y=\frac{-57}{-7}\\ =8\end{array}$

  $\begin{array}{c}z=\frac{35}{-5}\\ =-5\end{array}$

Thus, the solution of the system is $\left(4,8,-5\right)$ .