Q40. - Algebra 2

Question

Use Cramer’s Rule to solve the system of equations $3x+8y=28$ and $5x-7y=-55$.

A. $\left(3,5\right)$

B. $\left(-4,5\right)$

C. $\left(4,2\right)$

D. $\left(-5,4\right)$

Step-by-Step Solution

Answer

The correct option is B.

Step 1- Determine the value of x

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

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

Substitute 3 in place of a, 5 in place of c, 8 in place of b, $-7$ in place of d, 28 in place of e and $-55$ in place of f into $x=\frac{\left|\begin{array}{cc}e& b\\ f& d\end{array}\right|}{\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|}$and perform simplification of determinants to write the value of x.

$\begin{array}{c}x=\frac{\left|\begin{array}{cc}28& 8\\ -55& -7\end{array}\right|}{\left|\begin{array}{cc}3& 8\\ 5& -7\end{array}\right|}\\ =\frac{28\cdot \left(-7\right)-8\cdot \left(-55\right)}{3\cdot \left(-7\right)-8\cdot 5}\\ =\frac{244}{-61}\\ =-4\end{array}$

Step 2 – Determine the value of y

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

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

Substitute 3 in place of a, 5 in place of c, 8 in place of b, $-7$ in place of d, 28 in place of e and $-55$ in place of f into $y=\frac{\left|\begin{array}{cc}a& e\\ c& f\end{array}\right|}{\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|}$and perform simplification of determinants to write the value of y.

$\begin{array}{c}y=\frac{\left|\begin{array}{cc}3& 28\\ 5& -55\end{array}\right|}{\left|\begin{array}{cc}3& 8\\ 5& -7\end{array}\right|}\\ =\frac{3\cdot \left(-55\right)-\left(28\right)\cdot 5}{3\cdot \left(-7\right)-8\cdot 5}\\ =\frac{-305}{-61}\\ =5\end{array}$

$\begin{array}{c}y=\frac{\left|\begin{array}{cc}3& 28\\ 5& -55\end{array}\right|}{\left|\begin{array}{cc}3& 8\\ 5& -7\end{array}\right|}\\ =\frac{3\cdot \left(-55\right)-\left(28\right)\cdot 5}{3\cdot \left(-7\right)-8\cdot 5}\\ =\frac{-305}{-61}\\ =5\end{array}$

Step 3- Write the solution

The obtained value of x is $-4$ and y is 5 so the solution is $\left(-4,5\right)$ that is matching with option B.