Problem 24 - Introductory and Intermediate Algebra for College Students 4th

Question

(GRAPH CAN NOT COPY) The average yearly salary of an American whose final degree is a doctorate is 39 dollar 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 126 dollar thousand. Find the average yearly salary of Americans with each of these final degrees.

Step-by-Step Solution

Answer

The average annual salary of an American with a Bachelor's degree is 55 thousand dollars, and with a Doctorate degree is 71 thousand dollars.

Step 1: Formulate the equations

The first piece of information given translates as D = 2B - 39, where D is the salary of a Doctorate degree holder and B is the salary of a Bachelor's degree holder. The second piece of information translates to D + B = 126. These two equations can be used to solve the problem.

Step 2: Substitute and solve

Substitute the first equation (D = 2B - 39) into the second equation to find the value of B. It results in (2B - 39) + B = 126. Simplify this to 3B = 165.

Step 3: Find B

Divide both sides of the equation 3B = 165 by 3 to get B = 55. This means the average annual salary of an American with a Bachelor's degree is 55 thousand dollars.

Step 4: Find D

Substitute B = 55 into the first equation D = 2B - 39. Thus, D = 2*55 - 39 = 71. The average annual salary of an American with a Doctorate degree is 71 thousand dollars.

Key Concepts

Systems of Equations

In algebra, a system of equations is a set of more than one equation working together. They help us find values for variables that satisfy all the equations in the system. In problems like the one given, systems of equations are crucial because they allow us to solve for two variables at once. To break it down simply:

• The first piece of information we have is that the yearly salary of someone with a doctorate degree is \(39,000 less than twice the salary of someone with a bachelor's degree. Mathematically, this is written as: \( D = 2B - 39 \).
• The second piece of information is their combined salary is \)126,000, which is written as: \( D + B = 126 \).

The two equations form a system of equations:1. \( D = 2B - 39 \)2. \( D + B = 126 \)By substituting one equation into the other, we solve for one variable first, and then use that solution to find the other variable. It’s like solving a puzzle with more pieces! Remember, systems of equations are everywhere from calculating costs in budgeting to even planning schedules.

Degree Attainment

Degree attainment refers to the level of education that someone completes. In this exercise, we're focused on the attainment of a bachelor's degree versus a doctorate degree. Different degrees represent different levels of expertise, and this is often reflected in salary differences. Here’s a simple rundown:

• Bachelor's Degree: This is typically a four-year undergraduate degree. It's often the first degree students achieve after finishing secondary school. Salary expectations for bachelor's degree holders can vary widely depending on their field of study.
• Doctorate Degree: This is an advanced degree, usually requiring several years of study beyond a bachelor’s and even a master’s degree in some fields. It's the highest academic degree one can earn, often involving original research. Consequently, doctorate degree holders often have higher salary potential.

In our example, the exercise shows that doctorate holders earn more than bachelor's holders, a practical scenario due to the advanced skills and qualifications they possess. Understanding degree attainment is crucial as it can have a significant impact on career opportunities and earnings.

Average Salary Calculation

Calculating average salaries using algebraic equations helps us quantify the economic benefits offered by different levels of education. In our exercise, we use algebraic methods to determine the average salary based on degree attainment. Here’s how the calculations unfold:

• We started with the equation \( D = 2B - 39 \), showing the relationship between the salaries of doctorate and bachelor's holders.
• Using the second equation \( D + B = 126 \), we substituted \( D \) from the first equation into it. This gave us a single equation with only one variable: \( 3B = 165 \).
• Solving \( 3B = 165 \) for \( B \), we divided both sides by 3, leading to \( B = 55 \). This tells us the average salary for a bachelor's degree holder is \(55,000 per year.
• Next, we found \( D \) by substituting back \( B = 55 \) into \( D = 2B - 39 \), calculating \( D = 71 \). Hence, the average salary for a doctorate holder is \)71,000 per year.

These calculations show how algebra can be applied to real-world problems, helping us extract valuable information and make informed decisions based on educational attainment and potential earnings.