Problem 57 - College Algebra with Corequisite Support

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: $42,000, Office Manager: $40,000, Truck Driver: $36,000.

Step 1: Define the variables

Let's define variables for each co-worker'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 the equations

From the problem statement, we have three conditions:1. The sum of the warehouse manager and office manager's salaries is \( 82,000 \). This gives us the equation: \[ W + O = 82,000 \] 2. The office manager makes \( 4,000 \) more than the truck driver. This gives us the equation: \[ O = T + 4,000 \] 3. The sum of the warehouse manager and the truck driver's salaries is \( 78,000 \). This gives us the equation: \[ W + T = 78,000 \]

Step 3: Solve the system of equations

We now have a system of three equations:1. \( W + O = 82,000 \)2. \( O = T + 4,000 \)3. \( W + T = 78,000 \)Substitute equation 2 into equation 1:\[ W + (T + 4,000) = 82,000 \]This simplifies to:\[ W + T + 4,000 = 82,000 \]Subtract \( 4,000 \) from both sides:\[ W + T = 78,000 \]Now notice this is the same as equation 3, which means our system is consistent. We can use either equation, substitute back into equation 2 to find the exact salary values.

Step 4: Calculate Office Manager's Salary

From step 3, we confirmed \( W + T = 78,000 \). Since \( O = T + 4,000 \), substitute to find \( T \). From step 3, we had: \[ W + O = 82,000 \] Substitute \( O = T + 4,000 \) into this equation: \[ W + (T + 4,000) = 82,000 \] \[ W + T + 4,000 = 82,000 \] Solving for \( T \): \[ W + T = 78,000 \Rightarrow T = 78,000 - W \] Substitute \( T \) back in the equation \( O = T + 4,000 \) we have: \[ O = (78,000 - W) + 4,000 \Rightarrow O = 82,000 - W \]

Step 5: Calculate Salaries

Set equal expressions for \( W \) from both equations: Using equation 3 \( W = 78,000 - T \) and substituting \( T = 36,000 \): \[ W = 78,000 - 36,000 = 42,000 \] Now find \( O \) knowing \( O = 82,000 - W \): \[ O = 82,000 - 42,000 = 40,000 \]

Step 6: Verify Truck Driver's Salary

Now let's calculate \( T \): Using \( O = T + 4,000 \), substitute \( O = 40,000 \): \[ T = 40,000 - 4,000 = 36,000 \]

Step 7: Final check

Check consistency with given equations:1. \( W + O = 42,000 + 40,000 = 82,000 \)2. \( O = 36,000 + 4,000 = 40,000 \)3. \( W + T = 42,000 + 36,000 = 78,000 \)All equations are satisfied, confirming the solution is correct.

Key Concepts

linear equations

Linear equations form the foundation of solving many algebraic problems and are crucial in understanding systems of equations. A linear equation is an equation that makes a straight line when it is graphed. In simple terms, it describes a relationship where one variable depends on one or more other variables in a linear manner. The general form is often written as \( ax + b = 0 \), where \( a \) and \( b \) are constants.

When dealing with a system of linear equations, like in the coworker salary problem, each line represents a condition or relationship that must be true. For example, in our problem we had the following linear equations:

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

These equations represent straight-line relationships between the salaries of the warehouse manager, office manager, and truck driver. Understanding how to manipulate and solve these equations is key to finding out each coworker's salary. By substituting values and using basic algebra, one is able to solve these equations and learn the values of the unknowns involved.

problem solving

Problem solving in algebra involves several crucial steps: understanding the problem, translating it into mathematical language through equations, and systematically solving those equations. Our salary distribution problem demonstrates these steps clearly.

First, we need to "understand the problem" by identifying all relevant data. In this case, it includes how the salaries relate to each other.

• The combined salary of the warehouse and office managers is \( 82,000 \)
• The office manager earns \( 4,000 \) more than the truck driver
• The combined salary of the warehouse manager and the truck driver is \( 78,000 \)

Next, "translating the problem" means defining variables and writing these conditions as equations that reflect the situation. This is followed by "solving the equations," a skill involving substitution and sometimes elimination, where we refine the given conditions until we arrive at the unknown values which satisfy all conditions in the problem.

Finally, confirm the solution by checking if the found values satisfy all initial conditions, ensuring that no steps were missed or miscalculated. Effective problem solving also means reviewing the work and verifying the results, as was demonstrated when re-checking all the solutions in the original exercise.

algebraic expressions

Algebraic expressions are combinations of numbers, variables, and operations that represent a specific value or relationship. In the context of our exercise, all of the salaries can be expressed as algebraic expressions based on the given conditions.

To break it down:

• \( W + O = 82,000 \) expresses the sum of the warehouse and office manager's salaries.
• \( O = T + 4,000 \) shows that the office manager earns \( 4,000 \) more than the truck driver.
• \( W + T = 78,000 \) shows the sum of the warehouse manager and truck driver's salaries.

These expressions are manipulated to solve for unknowns. They showcase how algebra allows us to express real-world situations through mathematics.

Manipulating these algebraic expressions, like substituting one expression into another, enables problem solvers to solve for particular variables. For example, substituting \( O = T + 4,000 \) into \( W + O = 82,000 \) reduced complexity and allowed us to find specific numerical solutions. Understanding how to effectively manage and simplify algebraic expressions is essential for solving systems of equations and is a key component of algebraic education.