Chapter 8.5, Problem 95E

To determine

To solve: The system of equations using Cramer’s rule.

Answer to Problem 95E

The solution for the given system of equations $\{\begin{array}{c}-5x+3y=-14\\ 7x-2y=2\end{array}$ is $\left(-2,-8\right)$ .

Explanation of Solution

Given information:$$

  $\{\begin{array}{c}-5x+3y=-14\\ 7x-2y=2\end{array}$

Calculation:

Matrix of coefficients $M=\left(\begin{array}{cc}-5& 3\\ 7& -2\end{array}\right)$

Answers column is $\left(\begin{array}{c}-14\\ 2\end{array}\right)$

Replacing the $x-$ column values with answer column values,

  $M_x=\left(\begin{array}{cc}-14& 3\\ 2& -2\end{array}\right)$

Replacing the $y-$ column values with answer column values,

  $M_y=\left(\begin{array}{cc}-5& -14\\ 7& 2\end{array}\right)$

Finding the determinant of $M=\left(\begin{array}{cc}-5& 3\\ 7& -2\end{array}\right)$ ,

  $\begin{array}{c}\text{Determinant of}\left(\begin{array}{cc}-5& 3\\ 7& -2\end{array}\right)=\left(-5\right)\left(-2\right)-\left(3\right)\left(7\right)\\ D=-11\end{array}$

Finding the determinant of $M_x=\left(\begin{array}{cc}-14& 3\\ 2& -2\end{array}\right)$ ,

  $\begin{array}{c}\text{Determinant of}\left(\begin{array}{cc}-14& 3\\ 2& -2\end{array}\right)=\left(-14\right)\left(-2\right)-\left(3\right)\left(2\right)\\ D_x=22\end{array}$

Finding the determinant of $M_y=\left(\begin{array}{cc}-5& -14\\ 7& 2\end{array}\right)$ ,

  $\begin{array}{c}\text{Determinant of}\left(\begin{array}{cc}-5& -14\\ 7& 2\end{array}\right)=\left(-5\right)\left(2\right)-\left(-14\right)\left(7\right)\\ D_y=88\end{array}$

By using the Cramer’s Rule,

  $\begin{array}{c}x=\frac{D_x}{D}=\frac{22}{-11}=-2\\ y=\frac{D_y}{D}=\frac{88}{-11}=-8\end{array}$

Thus we can find the solutions for the given system of equations using Cramer’s Rule.