Q6. - Algebra 2

Question

Use Cramer’s Rule to solve each system of equations.

$\frac{1}{2}r-\frac{2}{3}s=2\frac{1}{3}$

$\frac{3}{5}r+\frac{4}{5}s=-10$

Step-by-Step Solution

Answer

The required solution is $\left(-6,-8\right)$.

Step 1- Determine the value of r

Apply the Cramer’s rule for the value of r. 

$r=\frac{\left|\begin{array}{cc}e& b\\ f& d\end{array}\right|}{\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|}$

Substitute $\frac{1}{2}$ in place of a, $\frac{3}{5}$ in place of c, $-\frac{2}{3}$ in place of b, $\frac{4}{5}$ in place of d, $2\frac{1}{3}$ in place of e and $-10$ in place of f into $r=\frac{\left|\begin{array}{cc}e& b\\ f& d\end{array}\right|}{\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|}$and perform simplification of determinants to write the value of r.

$\begin{array}{c}r=\frac{\left|\begin{array}{cc}2\frac{1}{3}& -\frac{2}{3}\\ -10& \frac{4}{5}\end{array}\right|}{\left|\begin{array}{cc}\frac{1}{2}& -\frac{2}{3}\\ \frac{3}{5}& \frac{4}{5}\end{array}\right|}\\ =\frac{\frac{7}{3}\cdot \frac{4}{5}-\left(-\frac{2}{3}\right)\cdot \left(-10\right)}{\frac{1}{2}\cdot \frac{4}{5}-\left(-\frac{2}{3}\right)\cdot \frac{3}{5}}\\ =\frac{-\frac{24}{5}}{\frac{4}{5}}\\ =-6\end{array}$

Step 2 – Determine the value of s

Apply the Cramer’s rule for the value of s. 

$s=\frac{\left|\begin{array}{cc}a& e\\ c& f\end{array}\right|}{\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|}$

Substitute $\frac{1}{2}$ in place of a, $\frac{3}{5}$ in place of c, $-\frac{2}{3}$ in place of b, $\frac{4}{5}$ in place of d, $2\frac{1}{3}$ in place of e and $-10$ in place of f into $s=\frac{\left|\begin{array}{cc}a& e\\ c& f\end{array}\right|}{\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|}$and perform simplification of determinants to write the value of s.

$\begin{array}{c}s=\frac{\left|\begin{array}{cc}\frac{1}{2}& 2\frac{1}{3}\\ \frac{3}{5}& -10\end{array}\right|}{\left|\begin{array}{cc}\frac{1}{2}& -\frac{2}{3}\\ \frac{3}{5}& \frac{4}{5}\end{array}\right|}\\ =\frac{\frac{1}{2}\cdot \left(-10\right)-\left(\frac{7}{3}\right)\cdot \left(\frac{3}{5}\right)}{\frac{1}{2}\cdot \frac{4}{5}-\left(-\frac{2}{3}\right)\cdot \frac{3}{5}}\\ =\frac{-\frac{32}{5}}{\frac{4}{5}}\\ =-8\end{array}$

Step 3- Write the solution

The obtained value of r is $-6$ and s is $-8$ so the solution is $\left(-6,-8\right)$.