Question

Write a system of equations and then solve for each variable.
3. The Arcadium arcade in Lynchburg, Tennessee uses 3 different colored tokens for their game machines. For
$20 you can purchase any of the following mixtures of tokens: 14 gold, 20 silver, and 24 bronze; OR, 20 gold, 15
silver, and 19 bronze; OR, 30 gold, 5 silver, and 13 bronze. What is the monetary value of each token?​

Answer 1

Answer:

Gold=$0.5

Silver=$0.35

Bronze=$0.25

Step-by-step explanation:

This is the system of equations:

$\ 20=14G+20S+24B$ (1)

$\ 20=20G+15S+19B$ (2)

$\ 20=30G+5S+13B$ (3)

Let's begin by substracting (2) from (1):

$\left \{ {{\ 20=14G+20S+24B} \atop {-\ 20=-20G-15S-19B}} \right$

$\ 0=-6G+5S+5B$ (4)

Isolating $G$ from (4):

$G=\frac{5S+5B}{6}$ (5)

Substituting (5) in (3):

$\ 20=30(\frac{5S+5B}{6})+5S+13B$

$\ 20=30S+38B$  (6)

Substracting (3) from (2):

$\left \{ {{\ 20=20G+15S+19B} \atop {-\ 20=-30G-5S-13B} \right$

$\ 0=-10G+10S=6B$

Isolating $G$:

$G=\frac{10S+6B}{10}$ (7)

Making (5)=(7):

$\frac{5S+5B}{6}=\frac{10S+6B}{10}$

Isolating $B$:

$B=\frac{5}{7}S$ (8)

Substituting (8) in (6):

$\ 20=30S+38(\frac{5}{7}S)$  

Isolating $S$:

$S=\ 0.35$  (9) This is the monetary value of silver token

Substituting (9) in (6):

$\ 20=30(\ 0.35)+38B$

Finding $B$:

$B=\ 0.25$ (10) This is the monetary value of bronze token

Substituting (10) and (9) in (1):

$\ 20=14G+20(\ 0.35)+24(\ 0.25)$

Finding $G$:

$G=\ 0.5$ (11) This is the monetary value of golden token