Questions: The Elentire Arcade uses three different colored tokens in their game machines. For 20 you can purchase any of the following combinations of tokens: 2 gold, 28 silver, and 40 bronze; 10 gold, 18 silver, and 10 bronze; or, 6 gold, 22 silver, and 30 bronze. Use x to represent the value of gold tokens, y to represent the value of silver tokens, and z to represent the value of bronze tokens. Write a system of three equations that directly models this situation. Do not substitute values in any equation while writing the system.

Solution Steps

To model the given situation, we need to set up a system of linear equations based on the value of each type of token. Each combination of tokens purchased for $20 gives us one equation. Let \( x \) be the value of a gold token, \( y \) be the value of a silver token, and \( z \) be the value of a bronze token.

The three combinations provided are:

1. 2 gold, 28 silver, and 40 bronze tokens.
2. 10 gold, 18 silver, and 10 bronze tokens.
3. 6 gold, 22 silver, and 30 bronze tokens.

From these combinations, we can write the following system of equations:

1. \( 2x + 28y + 40z = 20 \)
2. \( 10x + 18y + 10z = 20 \)
3. \( 6x + 22y + 30z = 20 \)

We have established a system of equations based on the combinations of tokens purchased for $20. The equations are as follows:

1. \( 2x + 28y + 40z = 20 \)
2. \( 10x + 18y + 10z = 20 \)
3. \( 6x + 22y + 30z = 20 \)

By solving the system, we find the values of the variables:

• \( x = 1 \)
• \( y = \frac{1}{2} \)
• \( z = \frac{1}{10} \)

The values represent the worth of each type of token:

• The value of a gold token \( x \) is \( 1 \).
• The value of a silver token \( y \) is \( \frac{1}{2} \).
• The value of a bronze token \( z \) is \( \frac{1}{10} \).

Final Answer