Q35. - Algebra 2

Question

For Exercises 34 and 35, use the following information.

An interior designer is preparing invoices for two of her clients. She has ordered silk dupioni and cotton damask fabric for both of them. 

Find the price per yard of each fabric.

Step-by-Step Solution

Answer

Price related to the silk is $35$ and the price related to the cotton is $25$. 

Step 1- Determine the value of x

Use the system of equations, 

$8x+13y=604.79$

$5\frac{1}{2}x+14y=542.30$

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 8 in place of a, $5\frac{1}{2}$ in place of c, 13 in place of b, 14 in place of d, $604.79$ in place of e and $542.30$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.

$\begin{array}{c}x=\frac{\left|\begin{array}{cc}604.79& 13\\ 542.4& 14\end{array}\right|}{\left|\begin{array}{cc}8& 13\\ \frac{11}{2}& 14\end{array}\right|}\\ =\frac{604.79\cdot 14-542.4\cdot 13}{8\cdot 14-\frac{11}{2}\cdot 13}\\ =\frac{1415.86}{\frac{81}{2}}\\ \approx 35\end{array}$ 

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 8 in place of a, $5\frac{1}{2}$ in place of c, 13 in place of b, 14 in place of d, $604.79$ in place of e and $542.30$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.

$\begin{array}{c}y=\frac{\left|\begin{array}{cc}8& 604.79\\ \frac{11}{2}& 542.4\end{array}\right|}{\left|\begin{array}{cc}8& 13\\ \frac{11}{2}& 14\end{array}\right|}\\ =\frac{8\cdot 542.4-\frac{11}{2}\cdot 604.79}{8\cdot 14-\frac{11}{2}\cdot 13}\\ =\frac{\frac{202571}{200}}{\frac{81}{2}}\\ \approx 25\end{array}$

Step 3- Write the solution

The obtained approximate value of x is 35 and y is 25 which mean that the price related to the silk is $35$ and the price related to the cotton is $25$.