Problem 34
Question
Finance The median player salary for the Dallas Cowboys was \(\$ 348,000\) in 2004 and \(\$ 555,000\) 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 2019 .
Step-by-Step Solution
Verified Answer
The linear equation for the median player salary for the Dallas Cowboys with respect to year is \(y = \$23,000x - \$44,892,000\). The predicted salary for the year 2019 is \$601,000.
1Step 1: Calculate the slope
The slope of a line is given by \((y2 - y1)/(x2 - x1)\). Here, \(y1 = \$348,000\) for the year \(x1 = 2004\), and \(y2 = \$555,000\) for the year \(x2 = 2013\). So the slope \(m\) is given by \((\$555,000 - \$348,000) / (2013 - 2004) = \$207,000 / 9 = \$23,000 per year.
2Step 2: Find the y-intercept
The slope-intercept form of a linear equation is \(y = mx + c\), where \(c\) is the y-intercept. We can substitute one of our given points and the determined slope into the equation to solve for \(c\). For simplicity, let's substitute \(x1\) and \(y1\), thus the equation is \$348,000 = $23,000 * 2004 + \(c\) . Solving for \(c\) gives \(c = \$348,000 - \$23,000 * 2004 = -\$44,892,000\).
3Step 3: Write the linear equation
Plugging \(m\) and \(c\) back into our equation, we find \(y = \$23,000x - \$44,892,000\). This is the equation for the median player salary for the Dallas Cowboys as a function of the year.
4Step 4: Predict the median salary in 2019
To predict the median salary in 2019 we substitute \(x = 2019\) in the equation, which gives \(y = \$23,000 * 2019 - \$44,892,000 = \$601,000\). So the predicted median player salary for the Dallas Cowboys in 2019 is \$601,000.
Key Concepts
Slope-Intercept FormMedian Salary PredictionSlope Calculation
Slope-Intercept Form
The slope-intercept form of a linear equation is a way to express any straight line using the formula \( y = mx + c \). Here, \( m \) represents the slope of the line, and \( c \) is the y-intercept, which is where the line crosses the y-axis. This method is particularly useful in identifying the relationship between two variables in a simple, straightforward way.
In our exercise, we used the slope-intercept form to establish a link between the year \( x \), as an independent variable, and the median salary \( y \), as a dependent variable. Once the slope \( m \) and the y-intercept \( c \) were determined, it became easy to express the formula that links the two factors over time.
In our exercise, we used the slope-intercept form to establish a link between the year \( x \), as an independent variable, and the median salary \( y \), as a dependent variable. Once the slope \( m \) and the y-intercept \( c \) were determined, it became easy to express the formula that links the two factors over time.
- \( m \) accounts for the rate of change in salary over time.
- \( c \) adjusts the equation to accommodate the starting value when \( x = 0 \).
Median Salary Prediction
By employing the linear equation obtained from the slope-intercept form, we can make predictions about the future median salaries. Once the equation \( y = 23000x - 44892000 \) is established, it becomes a powerful tool for forecasting.
To predict the median salary in any given year, simply replace \( x \) with the desired year and solve for \( y \). This exercise predicted the median salary in 2019 using this method:
However, remember that this is a simplified model and real-world factors may cause actual values to deviate.
To predict the median salary in any given year, simply replace \( x \) with the desired year and solve for \( y \). This exercise predicted the median salary in 2019 using this method:
- Plug 2019 into the equation: \( y = 23000 \times 2019 - 44892000 \).
- Carry out the math to find \( y = 601,000 \).
However, remember that this is a simplified model and real-world factors may cause actual values to deviate.
Slope Calculation
The slope of a line is a crucial component in understanding and calculating linear equations. Slope describes the steepness of the line, or how much \( y \) changes with a change in \( x \).
In our exercise, the slope \( m \) was found using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). With provided points for the years 2004 and 2013:
\( m = \frac{555,000 - 348,000}{2013 - 2004} = \frac{207,000}{9} = 23,000 \).
This means, on average, the salary increased by $23,000 each year, which is the rate of change or the slope of the line. This slope is integral to predicting future values within the constructed linear model.
In our exercise, the slope \( m \) was found using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). With provided points for the years 2004 and 2013:
- Starting point: 2004, salary \(348,000.
- Ending point: 2013, salary \)555,000.
\( m = \frac{555,000 - 348,000}{2013 - 2004} = \frac{207,000}{9} = 23,000 \).
This means, on average, the salary increased by $23,000 each year, which is the rate of change or the slope of the line. This slope is integral to predicting future values within the constructed linear model.
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