Problem 39 - Algebra and Trigonometry with Analytic Geometry

Question

A plumber and an electrician are each doing repairs on their offices and agree to swap services. The number of hours spent on each of the projects is shown in the following table. $$ \begin{array}{|l|c|c|} \hline & \begin{array}{c} \text { Plumber's } \\ \text { office } \end{array} & \begin{array}{c} \text { Electrician's } \\ \text { office } \end{array} \\ \hline \text { Plumber's hours } & 6 & 4 \\ \text { Electrician's hours } & 5 & 6 \\ \hline \end{array} $$ They would prefer to call the matter even, but because of tax laws, they must charge for all work performed. They agree to select hourly wage rates so that the bill on each project will match the income that each person would ordinarily receive for a comparable job. (a) If \(x\) and \(y\) denote the hourly wages of the plumber and electrician, respectively, show that $$ 6 x+5 y=10 x \text { and } 4 x+6 y=11 y . $$ Describe the solutions to this system. (b) If the plumber ordinarily makes \(\$ 35\) per hour, what should the electrician charge?

Step-by-Step Solution

Answer

The electrician should charge $28 per hour to satisfy the billing requirements.

Step 1: Understand the Problem

The problem involves determining the hourly wage rates of a plumber and an electrician such that the cost of provided services matches what they would ordinarily earn for comparable jobs. We must set up equations based on the hours worked and hourly rates.

Step 2: Set Up Equations

For the plumber to receive the equivalent of what he would earn for 6 hours in the electrician's office, we have the equation: \(6x + 5y = 10x\) (because 10 hours of the plumber’s rate is equivalent to his work in the plumber's office). Similarly, for the electrician to receive his usual earnings, we have: \(4x + 6y = 11y\).

Step 3: Simplify the Equations

Simplify each equation:1. From \(6x + 5y = 10x\), rearrange to get \(6x - 10x + 5y = 0\), which simplifies to \(-4x + 5y = 0\).2. From \(4x + 6y = 11y\), rearrange to get \(4x + 6y - 11y = 0\), which simplifies to \(4x - 5y = 0\).

Step 4: Solve the System of Equations

We now have the system of equations:1. \(-4x + 5y = 0\)2. \(4x - 5y = 0\)Notice these equations are equivalent, meaning every solution for one is a solution for the other. Solving the first, we find \(5y = 4x\), giving \(y = \frac{4}{5}x\). This means any wage combination where the electrician's rate is \(\frac{4}{5}\) of the plumber's rate is valid.

Step 5: Determine Electrician's Hourly Rate for Plumber's Rate $35

Since the plumber's hourly rate \(x\) is \(35, plug this value into the relation \(y = \frac{4}{5}x\).\(y = \frac{4}{5}\times 35 = 28\).Thus, the electrician should charge \)28 per hour.

Key Concepts

Hourly Wages

Hourly wages are the payments workers receive per hour for the labor they perform. In the context of this work exchange problem between a plumber and an electrician, understanding hourly wages becomes crucial.
Here, each professional has an hourly rate that both parties wish to respect to equitably exchange services. This ensures that the compensation for hours worked reflects what each would typically earn.

• The plumber's hourly wage is denoted by \(x\).
• The electrician's hourly wage is represented by \(y\).

By maintaining the integrity of these hourly wages, the plumber and electrician can engage in a fair exchange of services without monetary exchange, aligning with market norms and personal earnings.

Solving Equations

Solving equations involves finding values for variables that make the equations true. In this problem, the hourly wages \(x\) and \(y\) are the variables in question.
To determine the respective hourly wages, we set up two equations based on the service exchange:

• \(6x + 5y = 10x\), ensuring the plumber's compensation for work done matches the income for his effort.
• \(4x + 6y = 11y\), ensuring the electrician's earnings are equivalent to his usual.

Rewriting these equations helps in systematically finding the solution. Simplifying and rearranging lead to \(-4x + 5y = 0\) and \(4x - 5y = 0\), further easing the solving process. The primary goal is to balance each professional’s income for the services rendered.

Work Exchange Problem

The work exchange problem highlights an arrangement where two professionals trade services such that neither feels financially disadvantaged.
In this situation, both the plumber and electrician want to equalize their workloads by ensuring their charges align with what they typically earn. This creates a win-win scenario.

• The plumber works 6 hours in the electrician's office, and the electrician works 5 hours in the plumber's office.
• Both agree to set their hourly wages accordingly so that the total "due" from one offsets the earnings of the other.

To satisfy this condition, no money changes hands, but each respects the other's labor value, making it compliant with tax needs and fair monetarily.

Equivalent Equations

Equivalent equations are different equations that have the same solutions. In this exercise, both equations \(-4x + 5y = 0\) and \(4x - 5y = 0\) illustrate this concept. Though they appear distinct, they lead to the same relationship between \(x\) and \(y\).
The solution to both equations

• relies on the fact that \(5y = 4x\),
• resulting in \(y = \frac{4}{5}x\).

This insight simplifies the task of setting proper wage rates. Understanding how to manipulate equations to reveal these relationships is key in solving system of equations efficiently.