Part 1
A plumber charges a fixed rate plus an hourly rate to do house calls. The ordered pair ​(2​,65​) indicates total charges of $ 65 for a job taking 2 hours comma and the ordered pair left parenthesis 8 comma 155 right parenthesis indicates total charges of $ 155 for a job taking 8 hours.
​a) Write a linear equation giving the total​ charges, y, in terms of the number of hours x.
​b) Use the equation found in part a to find how much it costs for 6 hours.
​c) How many hours would result in a total charge of $ 140​?

Let's solve the problem step by step.

### Part a) Write a linear equation giving the total charges, $ y $, in terms of the number of hours $ x $.

We are given two points, $(x_1, y_1) = (2, 65)$ and $(x_2, y_2) = (8, 155)$. These points represent the total charges for a certain number of hours worked. The general form of a linear equation is:

\[
y = mx + b
\]

where:
- $ m $ is the slope of the line, and
- $ b $ is the y-intercept, representing the fixed rate (the charge when $ x = 0 $).

#### Step 1: Find the slope $ m $.

The slope $ m $ is calculated using the formula:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Substituting the given points:

\[
m = \frac{155 - 65}{8 - 2} = \frac{90}{6} = 15
\]

So, the slope $ m = 15 $.

#### Step 2: Find the y-intercept $ b $.

We use one of the points to solve for $ b $ using the equation $ y = mx + b $. Let's use the point $ (2, 65) $:

\[
65 = 15(2) + b
\]

\[
65 = 30 + b
\]

\[
b = 65 - 30 = 35
\]

So, the y-intercept $ b = 35 $.

#### Step 3: Write the equation.

The linear equation representing the total charges $ y $ in terms of the number of hours $ x $ is:

\[
y = 15x + 35
\]

### Part b) Use the equation found in part a to find how much it costs for 6 hours.

We substitute $ x = 6 $ into the equation:

\[
y = 15(6) + 35
\]

\[
y = 90 + 35 = 125
\]

So, it costs $125 for 6 hours.

### Part c) How many hours would result in a total charge of $140?

We set $ y = 140 $ and solve for $ x $:

\[
140 = 15x + 35
\]

Subtract 35 from both sides:

\[
105 = 15x
\]

Divide by 15:

\[
x = \frac{105}{15} = 7
\]

So, it would take 7 hours to result in a total charge of $140.

---

Would you like more details or have any further questions? Here are five related questions that could expand your understanding:

1. What is the meaning of the slope in the context of this problem?
2. How would the linear equation change if the hourly rate were different?
3. What is the significance of the y-intercept in real-world scenarios like this?
4. How would you interpret the equation graphically?
5. If the fixed rate increased by $10, how would that affect the total charges?

**Tip:** When solving for unknowns in linear equations, always check your work by substituting values back into the original equation to verify correctness.

Explore how to solve a linear equation problem involving total charges for plumbing services based on hours worked. Includes step-by-step solutions for finding the equation, calculating costs for specific hours, and determining hours for a given total charge. Suitable for Grades 7-9 students learning about linear equations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation