Chapter 3.7, Problem 39PPS

To determine

To solve: The system of linear equations $6x-5y=73$ and $-7x+3y=-71$ using Cramer’s rule.

Answer to Problem 39PPS

The solution set of the system of linear equations is $\left\{\left(8,-5\right)\right\}$ .

Explanation of Solution

Given:

The equations are $6x-5y=73$ and $-7x+3y=-71$ .

Concept used:

To solve the system of linear equations by Cramer’s rule for order $2\times 2$ :

The system of linear equation is given below:

  $\begin{array}{c}{a}_{1}x+b_{1}y=c_1\\ a_{2}x+b_{2}y=c_2\end{array}$

The solution of system is given by $x=\frac{D_x}{D}$ and $y=\frac{D_y}{D}$ , where $D_x=\det \left[\begin{array}{cc}{c}_1& b_1\\ c_2& b_2\end{array}\right]$ , $D_y=\det \left[\begin{array}{cc}{a}_1& c_1\\ a_2& c_2\end{array}\right]$ and $D=\det \left[\begin{array}{cc}{a}_1& b_1\\ a_2& b_2\end{array}\right]$ , $D$ is not equal to zero.

Cramer’s rule does not apply if $D=0$ , the system is inconsistent or has dependent equations.

Calculation:

Compare the first equation $6x-5y=73$ with the general form $a_{1}x+b_{1}y=c_1$ then the value of $a_1$ is $6$ , $b_1$ is $-5$ and $c_1$ is $73$ .

Compare the second equation $-7x+3y=-71$ with the general form $a_{2}x+b_{2}y=c_2$ then the value of $a_2$ is $-7$ , $b_2$ is $3$ and $c_2$ is $-71$ .

The value of $D$ is calculated as follows:

  $\begin{array}{c}D=\det \left[\begin{array}{cc}6& -5\\ -7& 3\end{array}\right]\\ =6\left(3\right)-\left(-7\right)\left(-5\right)\\ =18-35\\ =-17\end{array}$

Therefore, the value of $D$ is $-17$ .

Substitute $73$ for $c_1$ , $-5$ for $b_1$ , $-71$ for $c_2$ , $3$ for $b_2$ in equation $D_x=\det \left[\begin{array}{cc}{c}_1& b_1\\ c_2& b_2\end{array}\right]$ as follows:

  $\begin{array}{c}{D}_x=\det \left[\begin{array}{cc}73& -5\\ -71& 3\end{array}\right]\\ =\left(73\right)\left(3\right)-\left(-71\right)\left(-5\right)\\ =-136\end{array}$

Substitute $6$ for $a_1$ , $73$ for $c_1$ , $-7$ for $a_2$ , $-71$ for $c_2$ in equation $D_y=\det \left[\begin{array}{cc}{a}_1& c_1\\ a_2& c_2\end{array}\right]$ as follows:

  $\begin{array}{c}{D}_y=\det \left[\begin{array}{cc}6& 73\\ -7& -71\end{array}\right]\\ =\left(6\right)\left(-71\right)-\left(-7\right)\left(73\right)\\ =85\end{array}$

Substitute $-136$ for $D_x$ and $-17$ for $D$ in equation $x=\frac{D_x}{D}$ to find the value of $x$ as follows:

  $\begin{array}{c}x=\frac{-136}{-17}\\ =8\end{array}$

Substitute $85$ for $D_y$ and $-17$ for $D$ in equation $y=\frac{D_y}{D}$ to find the value of $y$ as follows:

  $\begin{array}{c}y=\frac{85}{-17}\\ =-5\end{array}$

Thus, the solution set of the system of linear equations is $\left\{\left(8,-5\right)\right\}$ .