Question

I'm studying for my algebra test and I need help with a problem involving linear equations. The temperature is 25 degrees Fahrenheit at 9:00 p.m. and is expected to drop 3 degrees as each hour passes. Can you find an explanation of how to write the linear equation that represents this scenario and what the y-intercept represents?.

Answer 1

Sure! Let's break down the scenario and write a linear equation to represent it.

1. **Identify the Variables**:
- Let \( T \) represent the temperature in degrees Fahrenheit.
- Let \( h \) represent the number of hours past 9:00 p.m.

2. **Initial Conditions**:
- At 9:00 p.m. (when \( h = 0 \)), the temperature is 25 degrees Fahrenheit. This is the initial value or the y-intercept of our equation.

3. **Rate of Change**:
- The temperature drops 3 degrees for each hour that passes. This means that the slope of our linear equation will be -3 (indicating a decrease).

4. **Writing the Linear Equation**:
A linear equation can be written in the form:
\[
T = mx + b
\]
Where:
- \( m \) is the slope,
- \( b \) is the y-intercept.

In our scenario:
- The slope \( m = -3 \) (temperature decreases by 3 degrees per hour).
- The y-intercept \( b = 25 \) (the temperature at 9:00 p.m.).

So we can write the linear equation as:
\[
T = -3h + 25
\]

5. **Understanding the Y-Intercept**:
The y-intercept (the value of \( T \) when \( h = 0 \)) represents the temperature at the starting point of the observation, which, in this case, is 9:00 p.m. Therefore, the y-intercept of 25 degrees Fahrenheit indicates that the initial temperature at 9:00 p.m. is 25 degrees.

In conclusion, the linear equation \( T = -3h + 25 \) describes how the temperature decreases over time, and the y-intercept represents the initial temperature at 9:00 p.m. Good luck with your algebra test!