Rose is going to hire a plumber to modify the layout of her bathroom. She wants to move the sink and the bathtub and has been told that the job will take 4 hours and will cost $414.00. If she only moves the sink, the job will take 1.5 hours and will cost $233.38.

Define the variables.
Write an equation to show the cost of hiring the plumber.
Rose decided that she also wanted to have the plumber change the faucet in the kitchen. This will mean the job will take 5 hours. What can she expect to pay?

could u use x, y and m? We're supposed to use one of the forms; slope intercept, general form or slope point

To tackle this problem, we will define the variables and then create equations that describe the cost of hiring the plumber. We can use $x$, $y$, and $m$ to represent the different aspects of the job and the cost.

**Step 1: Define the Variables**
- Let $x$ represent the time taken to complete the job in hours.
- Let $y$ represent the total cost of the job in dollars.
- Let $m$ represent the cost per hour of labor.

**Step 2: Write the Equations**

Given:
1. Moving the sink and the bathtub takes 4 hours and costs $414.00.
2. Moving only the sink takes 1.5 hours and costs $233.38.

We can assume that the plumber charges a fixed hourly rate ($m$) and a fixed fee ($b$) for each job.

Using the form $ y = mx + b $:

For the first scenario:
\[ 414 = 4m + b \]

For the second scenario:
\[ 233.38 = 1.5m + b \]

**Step 3: Solve the System of Equations**

We have:
\[ 414 = 4m + b \]
\[ 233.38 = 1.5m + b \]

We will solve these equations simultaneously to find $m$ and $b$.

Subtract the second equation from the first:
\[ 414 - 233.38 = 4m + b - (1.5m + b) \]
\[ 180.62 = 2.5m \]
\[ m = \frac{180.62}{2.5} \]
\[ m = 72.248 \]

Now substitute $m$ back into one of the original equations to find $b$:
\[ 233.38 = 1.5(72.248) + b \]
\[ 233.38 = 108.372 + b \]
\[ b = 233.38 - 108.372 \]
\[ b = 125.008 \]

So the hourly rate $m$ is $72.248, and the fixed fee $b$ is $125.008.

**Step 4: Write the Final Equation**

Using the slope-intercept form $ y = mx + b $:
\[ y = 72.248x + 125.008 \]

**Step 5: Calculate the Cost for 5 Hours**

If Rose decides to have the plumber change the faucet in the kitchen, making the total job time 5 hours, we can calculate the expected cost using the equation:

\[ y = 72.248(5) + 125.008 \]
\[ y = 361.24 + 125.008 \]
\[ y = 486.248 \]

Rose can expect to pay $486.25 (rounded to the nearest cent).

Would you like more details or have any questions? Here are 5 questions you can ask:

1. How would the cost change if the job took 6 hours instead of 5?
2. Can you explain how the system of equations was solved step by step?
3. What if the plumber had a different hourly rate? How would that affect the cost?
4. How can we check if our solution for $m$ and $b$ is correct?
5. Can this method be used for other types of cost calculations?

**Tip:** Always double-check your calculations and ensure all units are consistent throughout the problem.

Learn how to calculate costs for plumbing jobs involving moving a sink, bathtub, and changing a faucet. Solve using linear equations and the slope-intercept form. Suitable for high school students and anyone interested in cost estimation for home improvement projects.

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