Chapter 11, Problem 19CT

To determine

To calculate: The solution of the system of equations $\{\begin{array}{l}4x+3y=-23\\ 3x-5y=19\end{array}$ by using Cramer’s rule.

Answer to Problem 19CT

Solution:

The solution of the system of equations $\{\begin{array}{l}4x+3y=-23\\ 3x-5y=19\end{array}$ by using Cramer’s rule is $x=-2,y=-5$ or $\left(-2,-5\right)$.

Explanation of Solution

Given information:

The system of linear equations, $\{\begin{array}{l}4x+3y=-23\\ 3x-5y=19\end{array}$.

Formula used:

Crammers rule:

The solution of the system of equations $\{\begin{array}{l}ax+by=s\\ cx+dy=t\end{array}$ is given by,

$x=\frac{\left|\begin{array}{cc}s& b\\ t& d\end{array}\right|}{\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|},y=\frac{\left|\begin{array}{cc}a& s\\ c& t\end{array}\right|}{\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|}$ provided that $\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|=ad-bc\ne 0$.

Calculation:

Compare given system of equations $\{\begin{array}{l}x-2y=4\\ 3x+2y=4\end{array}$ with $\{\begin{array}{l}ax+by=s\\ cx+dy=t\end{array}$,

Gives, $a=4,b=3,c=3,d=-5,s=-23$ and $t=19$.

$\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|=\left|\begin{array}{cc}4& 3\\ 3& -5\end{array}\right|=-20-\left(9\right)=-29\ne 0$.

By using Cramer’s rule solutions are,

$x=\frac{\left|\begin{array}{cc}-23& 3\\ 19& -5\end{array}\right|}{\left|\begin{array}{cc}4& 3\\ 3& -5\end{array}\right|},y=\frac{\left|\begin{array}{cc}4& -23\\ 3& 19\end{array}\right|}{\left|\begin{array}{cc}4& 3\\ 3& -5\end{array}\right|}$

$\Rightarrow x=\frac{115-\left(57\right)}{-29},y=\frac{76-(-69)}{-29}$

$\Rightarrow x=\frac{58}{-29},y=\frac{145}{-29}$

$\Rightarrow x=-2,y=-5$

Thus, the solution of the system of equations $\{\begin{array}{l}x-2y=4\\ 3x+2y=4\end{array}$ by using Cramer’s rule is $x=-2,y=-5$ or using ordered pair $\left(-2,-5\right)$.