Problem 33 - Algebra and Trigonometry Real Mathematics, Real People

Question

Finance The median player salary for the New York Yankees was \(\$ 1.5\) million in 2007 and \(\$ 1.7\) million in 2013. Write a linear equation giving the median salary \(y\) in terms of the year \(x\). Then use the equation to predict the median salary in 2020 .

Step-by-Step Solution

Answer

The median player salary for the New York Yankees in 2020 would be approximately \$1.83333 million.

Step 1: Determine the Slope

First, it is necessary to calculate the slope of the linear relationship. The slope is the rate at which the median player salary is changing year by year. The slope formula in this context is (y2 - y1) / (x2 - x1). Here, (x1, y1) and (x2, y2) are two points on the salary line representing the year and corresponding median salary. Using the provided points (2007, \$1.5 million) and (2013, \$1.7 million), the slope becomes (1.7-1.5) / (2013-2007) = 0.033333 million/year or \$33,333/year.

Step 2: Form the Linear Equation

The linear function can be written using slope-intercept form y = mx + b. Here, m is the slope, x is the input year, and b is the y-intercept (the salary in the year when x=0). Substituting the slope (m) and one of the initial given points (2007, \$1.5 million) into this formula, we can solve for b as follows: 1.5 = 0.033333*2007 + b, which gives b = -66.167, or -$66.167 million. So, the linear function is y = 0.033333x - 66.167.

Step 3: Predict the Median Salary

Now we use obtained equation to predict the median salary in 2020. Plugging x = 2020 into the equation: y = 0.033333*(2020) - 66.167, gives y = 1.83333 or $1.83333 million.

Key Concepts

Slope Calculation

When we're dealing with any linear equation, understanding slope is crucial. The slope represents how much the dependent variable (in our case, the median salary) changes for a given change in the independent variable (year). You can think of it as the 'steepness' of the line.

To calculate the slope (\( m \)), we use the formula \( m = \frac{{y_2 - y_1}}{{x_2 - x_1}} \). This formula tells us:

• The difference in salaries (\( y_2 - y_1 \)) divided by the difference in years (\( x_2 - x_1 \)).

In our example, with the coordinates (2007, 1.5 million) and (2013, 1.7 million), the slope is \( m = \frac{{1.7 - 1.5}}{{2013 - 2007}} = 0.033333 \). This means the Yankees' median salary increased by approximately $33,333 each year.

Linear Function

A linear function is a simple mathematical way to describe a consistent relationship between two variables. Think of it like a formula that tells us how one variable affects another.

The general form of a linear equation is \( y = mx + b \), where:

• \( y \) is the value we want to find (median salary).
• \( m \) is the slope we calculated before.
• \( x \) is the year.
• \( b \) is the y-intercept, or the predicted salary when the year is zero.

In our example, substituting known values, we find \( b \) using the slope and one point: \( 1.5 = 0.033333 \times 2007 + b \). Solving gives \( b = -66.167 \). Thus, our function is \( y = 0.033333x - 66.167 \), capturing the linear relationship between year and median salary.

Predicting Future Values

One of the most exciting aspects of linear functions is their use in predicting future outcomes. Once we have our linear equation, we can estimate future values by simply substituting future dates into the equation.

For instance, to predict the median salary in 2020, we substitute \( x = 2020 \) in our equation:

• \( y = 0.033333 \times 2020 - 66.167 \)
• This calculation yields \( y \approx 1.83333 \) million.

Therefore, using our linear model, the predicted median salary in 2020 would be about $1.833 million. This method allows us to estimate trends and make informed decisions based on past data.

Median Salary Analysis

In finance, analyzing median salaries is important as it provides insight into typical earnings, reducing the impact of outliers. The median represents the middle point where half of the data lies above and half below. This makes it a robust tool in understanding salaries.

By using linear equations, we can't just track the rate of change, but also project future salary trends. Such projections can inform sports teams, businesses, and policymakers in their planning and decision-making. Considerations include:

• Understanding how inflation and other market variables might impact salaries.
• Acknowledging that linear models are simplifications and might not capture complex real-world nuances fully.

Thus, while linear equations provide valuable predictions, always consider the broader financial context and other influencing factors.