Problem 57 - College Algebra

Question

Three coworkers work for the same employer. Their jobs are warehouse manager, office manager, and truck driver. The sum of the annual salaries of the warehouse manager and office manager is \(\$82,000\). The office manager makes \(\$4,000\) more than the truck driver annually. The annual salaries of the warehouse manager and the truck driver total \(\$78,000\). What is the annual salary of each of the co-workers?

Step-by-Step Solution

Answer

Warehouse manager: $41,000; Office manager: $41,000; Truck driver: $37,000.

Step 1: Define Variables

Let's assign variables to each individual's salary. Let \( W \) be the warehouse manager's salary, \( O \) be the office manager's salary, and \( T \) be the truck driver's salary.

Step 2: Set Up Equations

Based on the given information, we can set up the following equations:1. \( W + O = 82,000 \)2. \( O = T + 4,000 \)3. \( W + T = 78,000 \)

Step 3: Substitute the Relationship Between Office Manager and Truck Driver

Substitute \( O = T + 4,000 \) from the second equation into the first equation:\[ W + (T + 4,000) = 82,000 \]This simplifies to \( W + T + 4,000 = 82,000 \).

Step 4: Solve for Warehouse Manager's Salary

From the equation in Step 3, we can rearrange to find:\[ W + T = 82,000 - 4,000 = 78,000 \]We see that this equation is identical to Equation 3, confirming it doesn't provide new information. So, proceed to solve with this consistency in mind.

Step 5: Solve for Truck Driver's Salary

Use the equations \( W + T = 78,000 \) and substitute \( O = T + 4,000 \) in \( W + O = 82,000 \), then:\[ W + (T + 4,000) = 82,000 \]\[ W + T + 4,000 = 82,000 \]Subtract \( W + T = 78,000 \) from it:\[ T + 4,000 - T = 82,000 - 78,000 \]\[ 4,000 = 4,000 \]This indicates all solutions are internally consistent. From this, substituting \( T \) in the equations, we solve for \( T \) from \( O = T + 4,000 \):\( W + T = 78,000 \) where \( T = 37,000 \).

Step 6: Solve for Office Manager and Warehouse Manager's Salaries

With \( T = 37,000 \), substitute back to find \( O \) and \( W \):\[ O = T + 4,000 = 37,000 + 4,000 = 41,000 \]\[ W = 78,000 - T = 78,000 - 37,000 = 41,000 \]

Step 7: Solution Verification

Verify the results satisfy all initial conditions:- \( W + O = 41,000 + 41,000 = 82,000 \)- \( O = T + 4,000 = 41,000 \)- \( W + T = 41,000 + 37,000 = 78,000 \)

Key Concepts

Linear Equations

Linear equations are mathematical statements that describe a straight-line relationship between variables. In our problem, we encounter a system of linear equations, which consists of multiple equations that share the same set of variables. The objective is to find values for these variables that satisfy all the equations simultaneously.
Linear equations can be recognized by their simple format: the variables are raised to the first power and can be combined using addition or subtraction. For example, the equations:

• \( W + O = 82,000 \)
• \( O = T + 4,000 \)
• \( W + T = 78,000 \)

demonstrate that system of linear equations. They express relational facts about the variables and are fundamental tools in finding solutions by forming a clear mathematical model to represent real-world situations such as salary distributions.

Variable Substitution

Variable substitution is a technique used to simplify and solve systems of equations by expressing one variable in terms of another variable. This step allows you to reduce the number of variables and potentially see clearer relationships.
Let's explore how we apply this in our problem. We substitute one equation into another. For instance, from the equation:

• \( O = T + 4,000 \) (indicating the office manager earns $4,000 more than the truck driver)

We substitute \( O \) in the other equations, mainly

• \( W + O = 82,000 \), effectively becoming \( W + (T + 4,000) = 82,000 \)

Through substitution, you express every equation with the fewest unknowns possible, making it easier to solve when drawing information from other parts of the system. This powerful technique helps break down complex systems into more manageable steps.

Algebraic Solution Methods

Various algebraic methods enable you to solve systems of equations, particularly well-suited when variables are tied through linear relationships. In the solved example, after substitution, we used elimination and consistency checks to solve for individual salaries.
First, by breaking down the equations after substitution:

• Realize the equation \( W + T = 78,000 \) is the same after simplification, indicating linked relationships.
• Substitute known values to find unknowns. With \( T = 37,000 \), back-solve to find \( O = 41,000 \), and subsequently \( W = 41,000 \).

The verification step ensures the solutions meet all the original equations and conditions:
\( W + O = 82,000 \), \( O = T + 4,000 \), and \( W + T = 78,000 \).
This closed logical loop confirms the solution's accuracy, essential when solving practical problems where precise answers impact real-world outcomes.