Q33. - Algebra 2

Question

For Exercises 32 and 33, use the following information.

Marcus purchases a game card to play virtual games at the arcade. His favorite games are the race car simulator, which costs 7 points for each play, and the snowboard simulator, which costs 5 points for each play. Marcus came with enough money to buy a 50-point card, and he has time to play 8 games.

Solve the system using Cramer’s Rule to find the number of times Marcus can play race car simulator and snowboard simulator.

Step-by-Step Solution

Answer

The count related to the race car is 5 and the count related to the snowboard is 3. 

Step 1- Determine the value of x

Use the system of equations, 

$7r+5s=50$

$5x+5y=40$

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 7 in place of a, 5 in place of c, 8 in place of b, 1 in place of d, $297.50$ in place of e and 4000 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.

$\begin{array}{c}x=\frac{\left|\begin{array}{cc}50& 5\\ 40& 5\end{array}\right|}{\left|\begin{array}{cc}7& 5\\ 5& 5\end{array}\right|}\\ =\frac{50\cdot 5-40\cdot 5}{7\cdot 5-5\cdot 5}\\ =\frac{50}{10}\\ =5\end{array}$

 

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 7 in place of a, 5 in place of c, 8 in place of b, 1 in place of d, $297.50$ in place of e and 4000 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.

$\begin{array}{c}y=\frac{\left|\begin{array}{cc}7& 50\\ 5& 40\end{array}\right|}{\left|\begin{array}{cc}7& 5\\ 5& 5\end{array}\right|}\\ =\frac{7\cdot 40-5\cdot 50}{7\cdot 5-5\cdot 5}\\ =\frac{30}{10}\\ =3\end{array}$

Step 3- Write the solution

The obtained value of x is 5 and y is 3 which mean that the count related to the race car is 5 and the count related to the snowboard is 3.