Use Linear Models
to Solve Problems
Learn to interpret slope and y-intercept in real-world linear models, and use the equation \( y = mx + b \) to make predictions from data.
What Is a Linear Model?
A linear model is an equation in the form \( y = mx + b \) that best describes the relationship between two variables in a data set. It lets us summarize a trend and make predictions beyond the data we collected.
From Data to a Line of Best Fit
Why Do Linear Models Matter?
Real-world data is messy. A linear model gives us a simple equation that captures the overall pattern so we can:
The Equation \( y = mx + b \)
Interpreting Slope in Context
The slope \( m \) always has units. To interpret it, say: "For each additional 1 [unit of \( x \)], \( y \) increases (or decreases) by \( m \) [units of \( y \)]."
Interpreting the Y-Intercept
The y-intercept \( b \) is the predicted value of \( y \) when \( x = 0 \). Ask yourself: "Does \( x = 0 \) make sense in this context?"
Slope vs. Y-Intercept
| Slope \( m \) | Y-Intercept \( b \) | |
|---|---|---|
| Meaning | Rate of change | Starting value |
| Question it answers | "How fast is \( y \) changing?" | "Where does \( y \) begin?" |
| Units | \( y \)-units per \( x \)-unit | Same units as \( y \) |
| On the graph | Steepness of the line | Where line crosses y-axis |
| Example | 1.5 cm/hr | 1 cm |
How to Apply a Linear Model
Example 1: Plant Growth
A biology experiment models plant height as \( y = 1.5x + 1 \), where \( x \) = hours of sunlight per day and \( y \) = mature plant height in cm. Predict the height with 6 hours of sunlight.
Example 2: Taxi Fare
A taxi company charges a fare modeled by \( y = 2.50x + 3 \), where \( x \) = miles driven and \( y \) = total fare in dollars. How much does a 12-mile ride cost?
Example 3: Temperature & Elevation
Hikers record temperature at different elevations. A linear model gives \( y = -3.5x + 80 \), where \( x \) = elevation in thousands of feet and \( y \) = temperature in °F. What does the slope mean? Predict the temperature at 6,000 ft.
Interpolation vs. Extrapolation
Common Mistakes to Avoid
Summary Cheat Sheet
| Concept | What to Do | Example |
|---|---|---|
| Interpret slope | "For each 1-unit increase in \( x \), \( y \) changes by \( m \)" | 1.5 cm per hr |
| Interpret intercept | "When \( x = 0 \), predicted \( y = b \)" | 1 cm at 0 hrs |
| Make a prediction | Substitute \( x \) into \( y = mx + b \) | \( y = 1.5(6) + 1 = 10 \) |
| Negative slope | "\( y \) decreases by \( |m| \) per unit" | -3.5°F per 1,000 ft |
| Check reasonableness | Is \( x \) within the data range? | Interpolation > Extrapolation |
Question 1 of 5
A linear model for a study on exercise is \( y = 0.8x + 2 \), where \( x \) = minutes of exercise per day and \( y \) = calories burned (in hundreds). What does the slope 0.8 mean?
Question 2 of 5
Using the model \( y = -2x + 50 \), where \( x \) = weeks and \( y \) = pounds of supplies remaining, predict the supplies remaining after 10 weeks.
Question 3 of 5
In the model \( y = 4.2x + 15 \), the y-intercept is 15. If \( x \) = years of experience and \( y \) = hourly wage in dollars, what does the 15 represent?
Question 4 of 5
Data was collected for students' study hours (\( x \)) and test scores (\( y \)) between 1 and 8 hours of studying. The model is \( y = 5x + 45 \). A student asks: "What score would I get if I studied 20 hours?" What is the best response?
Question 5 of 5
Two linear models describe tree growth:
Tree A: \( y = 2x + 5 \)
Tree B: \( y = 3x + 1 \)
where \( x \) = years and \( y \) = height in feet. Which tree started taller, and which grows faster?