Challenge The value of a baseball​ player's rookie card began to increase once the player retired. When he retired in 1998 his card was worth ​$7.25. The value has increased by ​$1.63 each year since then. Express the relationship relating the value of the card y in dollars and the number of years x the player has been in retirement with an equation. Is the relationship between x and y​ proportional? What was the value of the card in 2006​?

Question

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Part 1
Express the relationship with an equation.
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enter your response here ​(Use integers or decimals for any numbers in the​ expression.)

Answer 1

To express the relationship relating the value of the baseball player's rookie card \( y \) in dollars with the number of years \( x \) since the player retired, we can establish the following linear equation:

1. The initial value of the card when the player retired in 1998 was \( 7.25 \) dollars.
2. The card value increases by \( 1.63 \) dollars each year.

Using these two pieces of information, we can write the equation as follows:

\[ y = 7.25 + 1.63x \]

Where:

• \( y \) is the value of the card in dollars.
• \( x \) is the number of years since the player retired.

Part 2: Proportional Relationship

The relationship is not proportional. A proportional relationship would require the equation to be of the form \( y = kx \) (where \( k \) is a constant), which means the line would go through the origin (0,0). Since the equation has a constant term of \( 7.25 \), it represents a linear function that does not pass through the origin.

Part 3: Value of the Card in 2006

To find the value of the card in 2006, we need to determine how many years it was from 1998 to 2006:

\[ 2006 - 1998 = 8 \text{ years} \]

Now, we can plug \( x = 8 \) into our equation:

\[ y = 7.25 + 1.63(8) \] \[ y = 7.25 + 13.04 \] \[ y = 20.29 \]

Thus, the value of the card in 2006 was $20.29.