One equation. Predict the future.
A linear model is just slope plus a starting point. It's everything you've learned in this topic, packed into a single line of math, and it's how a number becomes a forecast.
You're saving up to study abroad next summer. Today you have $200 in your account, and you save about $75 per week. Your friend asks if you'll have enough by week 24.
You don't need a calculator to answer. You have a starting amount, you have a rate of change, and you have a future moment you want to peek into. Put those together and you've built a linear model: the most useful piece of math in any 100-level course. By the end of this lesson, you'll write that equation in your sleep, and use it to predict everything from your grocery bill to your graduation savings.
Slope tells you how. The intercept tells you where you start.
The y-intercept is where the line crosses the y-axis. your starting point. The slope is how steeply it climbs (or descends) from there.
The slope-intercept form. m is the slope, b is the y-intercept.
A linear model is an equation in the form y = mx + b that describes a real-world quantity changing at a constant rate.
Prediction uses the model to find an unknown y by plugging in any x you want, including ones you haven't observed yet.
The vocabulary you actually need
- Linear model An equation that describes how one quantity changes at a constant rate as another quantity changes. Always takes the form y = mx + b.
- Slope-intercept form The format y = mx + b. It's called this because m is the slope and b is the y-intercept, both ingredients are right there in the equation, no rearranging needed.
- y-intercept (b) Where the line crosses the y-axis, the value of y when x = 0. In word problems, it's the starting amount, base fee, or fixed cost.
- Independent variable The input. The thing you control or that "drives" the change. Goes on the x-axis.
- Dependent variable The output. The thing that depends on the input. Goes on the y-axis.
- Extrapolation Using your model to predict beyond your data. Powerful, but also where models go wrong if the underlying pattern changes.
Building a savings model, start to finish.
"You currently have $200 saved. You save $75 per week. Build a model for your savings, then predict how much you'll have at week 24, and decide whether you'll hit $1,800 in time."
Identify the slope and the y-intercept.
Read the situation slowly. The per word tells you the slope. The starting amount is the y-intercept, what's true when zero weeks have passed.
| Phrase | What it means |
|---|---|
| "$75 per week" | m = 75 (slope, dollars per week) |
| "$200 today" | b = 200 (y-intercept, starting savings) |
Plug them into y = mx + b.
Now it's just substitution. The slope replaces m; the y-intercept replaces b.
Sanity-check before predicting.
Try a value you can verify. At x = 0 (today), the model predicts y = 75(0) + 200 = $200. ✓ matches your current savings. At x = 1 (next week), y = 75(1) + 200 = $275, that's $200 plus one week of $75. ✓ Both check out, so the model is set up correctly.
Predict at week 24.
Plug x = 24 into the model.
You'll have $2,000 by week 24, comfortably past your $1,800 goal. In fact, you can flip the question: solving 1,800 = 75x + 200 gives x ≈ 21.3 weeks, so you'll cross the finish line in about week 22.
Three problems. Build, predict, identify.
A streaming service charges $12 per month, plus a one-time $20 sign-up fee. Let x = months and y = total cost. Fill in the slope and y-intercept.
A car rental costs y = 0.25x + 45, where x is miles driven and y is total dollars. What's the cost for a 200-mile trip?
A small online store models its monthly profit as y = 8x − 300, where x is the number of items sold. What does the −300 tell you?
Three fast questions before we move on.
Q1. A gym membership is modeled by y = 35x + 60, where x is months and y is total dollars paid. What does 60 represent?
Q2. A water tank starts with 50 gallons and drains at 2 gallons per minute. Which equation models the gallons remaining (y) after x minutes?
Q3. A linear model predicts a city's population at y = 1,500x + 80,000, where x is years since 2025. According to the model, what will the population be in 2035?
Now let's make Excel do all of this for you.
You can build a linear model by hand. You can predict any value you want. But what if you have 50 data points instead of 2? What if the input changes every day and you want the prediction to update automatically? That's where spreadsheets earn their keep. In Lesson 6, we put everything from Topic 1 (functions, slope, models, predictions) into Excel, and watch the math we just learned do its job at scale.
Continue to Lesson 06Different angle? Need another rep? These are optional — tap any that look helpful.