Challenge The value of a baseball​ player's rookie card began to increase once the player retired. When he retired in 1997 his card was worth ​$4.38. The value has increased by ​$2.55 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 2008​?

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 between the value of the baseball player's rookie card \(y\) in dollars and the number of years \(x\) the player has been in retirement since 1997, we can use the following linear equation:

\[ y = 4.38 + 2.55x \]

Here, \(4.38\) is the initial value of the card when the player retired in 1997, and \(2.55\) is the amount the value increases each year.

Part 2: Is the relationship between \(x\) and \(y\) proportional?

A relationship is proportional if it can be expressed in the form \(y = kx\), where \(k\) is a constant. In this case, the equation we have includes a fixed starting value (\(4.38\)) as well as a changing value based on \(x\). Since there is a constant term (the \(4.38\)), the relationship is not proportional.

Part 3: What was the value of the card in 2008?

To find the value of the card in 2008, we first determine how many years have passed since the player retired in 1997.

• The year 2008 is \(2008 - 1997 = 11\) years after 1997. Thus, \(x = 11\).

Now we can substitute \(x = 11\) into the equation:

\[ y = 4.38 + 2.55(11) \] \[ y = 4.38 + 28.05 \] \[ y = 32.43 \]

So, the value of the card in 2008 was $32.43.