Q37. - Algebra 2

Question

For Exercises 36 and 37, use the following information.

The Harvest Nut Company sells made-to-order tail mixes. Santito’s favourite mix contains peanuts, raisin, and carob-coated pretzels. Peanuts sell for $3.20$ per pound, raisins are $2.40$ per pound, and the carob-coated pretzels are $4.00$ per pound. Santito chooses to have twice as many pounds of pretzels as raisins, wants 5 pounds of mix, and can afford $16.80$.

How many pounds of peanuts, raisins, and carob-coated pretzels can Santito buy?

Step-by-Step Solution

Answer

The amount of peanuts is 2 pounds, amount of raisins is 1 pound and carob-coated pretzels is 2 pounds.

Step 1- Determine the value of x

Use the following system of equations. 

$x+y+z=5$

$z=2y$

$3.2x+2.4y+4z=16.80$

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

$x=\frac{\left|\begin{array}{ccc}j& b& c\\ k& e& f\\ l& h& i\end{array}\right|}{\left|\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right|}$

Use the equations and substitute the value of a, b, c, d, e, f, g, h, i, j, k and l into $x=\frac{\left|\begin{array}{ccc}j& b& c\\ k& e& f\\ l& h& i\end{array}\right|}{\left|\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right|}$after it perform simplification of determinants to write the value of x.

$\begin{array}{c}x=\frac{\left|\begin{array}{ccc}5& 1& 1\\ 0& -2& 1\\ 16.8& 2.4& 4\end{array}\right|}{\left|\begin{array}{ccc}1& 1& 1\\ 0& -2& 1\\ 3.2& 2.4& 4\end{array}\right|}\\ =\frac{-\frac{8}{5}}{-\frac{4}{5}}\\ =2\end{array}$

Step 2 – Determine the value of y

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

$y=\frac{\left|\begin{array}{ccc}a& j& c\\ d& k& f\\ g& l& i\end{array}\right|}{\left|\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right|}$

Use the equations and substitute the value of a, b, c, d, e, f, g, h, i, j, k and l into $y=\frac{\left|\begin{array}{ccc}a& j& c\\ d& k& f\\ g& l& i\end{array}\right|}{\left|\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right|}$after it perform simplification of determinants to write the value of y.

$\begin{array}{c}y=\frac{\left|\begin{array}{ccc}1& 5& 1\\ 0& 0& 1\\ 3.2& 16.8& 4\end{array}\right|}{\left|\begin{array}{ccc}1& 1& 1\\ 0& -2& 1\\ 3.2& 2.4& 4\end{array}\right|}\\ =\frac{-\frac{4}{5}}{-\frac{4}{5}}\\ =1\end{array}$

Step 3 – Determine the value of z

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

$z=\frac{\left|\begin{array}{ccc}a& b& j\\ d& e& k\\ g& h& l\end{array}\right|}{\left|\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right|}$

Use the equations and substitute the value of a, b, c, d, e, f, g, h, i, j, k and l into $z=\frac{\left|\begin{array}{ccc}a& b& j\\ d& e& k\\ g& h& l\end{array}\right|}{\left|\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right|}$after it perform simplification of determinants to write the value of z.

$\begin{array}{c}z=\frac{\left|\begin{array}{ccc}1& 1& 5\\ 0& -2& 0\\ 3.2& 2.4& 16.8\end{array}\right|}{\left|\begin{array}{ccc}1& 1& 1\\ 0& -2& 1\\ 3.2& 2.4& 4\end{array}\right|}\\ =\frac{-\frac{8}{5}}{-\frac{4}{5}}\\ =2\end{array}$

Step 4- Write the solution

The obtained value of x is 2, y is 1 and z is 2 so the amount of peanuts is 2 pounds, amount of raisins is 1 pound and carob-coated pretzels is 2 pounds.