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

Gold=$0.5

Silver=$0.35

Bronze=$0.25

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=(5S+5B)/(6)` (5)

Substituting (5) in (3):

`\$20=30((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=(10S+6B)/(10)` (7)

Making (5)=(7):

`(5S+5B)/(6)=(10S+6B)/(10)`

Isolating
`B`:

`B=(5)/(7)S` (8)

Substituting (8) in (6):

`\$20=30S+38((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