Chapter 4.5, Problem 4BCYP

To determine

To find: The value of $x$ , $y$ in the given system by using Cramer’s rule.

Answer to Problem 4BCYP

The value of $x$ is $5$ and $y$ is $-6$ .

Explanation of Solution

Given information:

The given system is:

  $\begin{array}{c}8x-5y=70\\ 9x+7y=3\end{array}$

Concept used:

The value of $x$ , $y$ in the given system by using Cramer’s rule is given as:

  $\begin{array}{c}x=\frac{D_x}{D}\\ y=\frac{D_y}{D}\end{array}$

Here, $D_x$ , $D_y$ are the determinant formed by replacing the $x$ column, $y$ column value with the answer column.

Here, $D$ is the determinant of the coefficient matrix of the given system.

The value of determinant $D$ , $D_x$ , $D_y$ is calculated as:

  $\begin{array}{c}D=\left|\begin{array}{cc}8& -5\\ 9& 7\end{array}\right|\\ =56+45\\ =101\end{array}$

Further simplified as:

  $\begin{array}{c}{D}_x=\left|\begin{array}{cc}70& -5\\ 3& 7\end{array}\right|\\ =490+15\\ =505\end{array}$

Further simplified as:

  $\begin{array}{c}{D}_y=\left|\begin{array}{cc}8& 70\\ 9& 3\end{array}\right|\\ =24-630\\ =-606\end{array}$

Using the Cramer’s rule, the value of $x$ , $y$ in the given system is calculated as:

  $\begin{array}{c}x=\frac{505}{101}\\ =5\\ y=\frac{-606}{101}\\ =-6\end{array}$

Therefore, the value of $x$ is $5$ and $y$ is $-6$ .