Problem 4 - Algebra and Trigonometry

Question

The median yearly salary of an American whose final degree is a doctorate is \(\$ 45\) thousand less than twice that of an American whose final degree is a bachelor's. Combined, two people with each of these educational attainments earn \(\$ 198\) thousand. Find the median yearly salary of Americans with each of these final degrees.

Step-by-Step Solution

Answer

The median yearly salary of an American with a bachelor's degree is \$81 thousand and that of an American with a doctorate is \$117 thousand.

Step 1: Represent the information as equations

Let \(x\) be the median yearly salary of an American with a bachelor’s degree, and let \(y\) be the median yearly salary of an American with a doctorate. We can set up two equations from the problem: Equation (1) from the first piece of information would be: \(y = 2x - 45\) Equation (2) from the second piece of information would be: \(x + y = 198\)

Step 2: Substitute equation (1) into equation (2)

Substitute equation (1) into equation (2) to solve for \(x\): \( x + 2x - 45 = 198\) which simplifies to \(3x - 45 = 198\)

Step 3: Solve for x

Add 45 to both sides of the equation and then divide by 3 to solve for \(x\): \(3x = 243\), thus \(x = 81\)

Step 4: Solve for y using x = 81 in equation (1)

Substitute \(x = 81\) into equation (1) to find the value of \(y\): \(y = 2(81) - 45\), which simplifies to \(y = 117\)

Key Concepts

Linear Equations

In algebra, a linear equation represents a straight line when graphed on a coordinate plane. More essentially, it's an equation that describes a relationship between variables using a constant rate of change.

For instance, in the given problem, the linear equations are used to relate the salaries of two different educational levels. We are given:

• The salary of a person with a bachelor's degree is denoted as \(x\).
• The salary of a person with a doctorate is denoted as \(y\).

From the problem, we derive the equations:- \(y = 2x - 45\) (This explains how the doctorate salary is related to the bachelor's salary.)- \(x + y = 198\) (This represents the combined salary from both degrees.)
An essential feature of linear equations is their predictability - they maintain a constant rate of change, which helps us relate and find unknown values efficiently.

Median Salary Calculation

The median salary is a statistical measure representing the middle value in a list of salaries. It's significant because it is not affected by extremely high or low salaries, unlike the average.

In the context of this problem, we're tasked with finding the median salary for individuals with bachelor's and doctorate degrees. The median gives us a better understanding of what a 'typical' salary might be for these educational levels.

• According to the problem, the doctorate salary is transformed by a linear relationship to the bachelor's salary.
• The combined salary for both educational levels helps confirm the accuracy of our calculations.

The use of median rather than average here indicates that we're looking for central tendencies, which provide a clearer picture of salary distribution without skewed results from extreme figures.

Substitution Method

The substitution method is a reliable algebraic technique to solve systems of equations, particularly when dealing with two variables. It involves substituting one variable in terms of another to simplify the problem.

Here's how it works in this problem:

• First, solve one equation for one variable. The equation \(y = 2x - 45\) gives \(y\) in terms of \(x\).
• Next, substitute this expression for \(y\) in the other equation, \(x + y = 198\).
• By substituting, we have a new equation: \(x + 2x - 45 = 198\).
• This equation simplifies to \(3x - 45 = 198\), and solving it gives the value of \(x\).
• Finally, substitute back the value of \(x\) to find \(y\).

This method is efficient because it reduces the variables step-by-step, making the solution easier to understand and compute.