A linear model fits a situation with a constant rate of change: a fixed fee plus a per-unit charge.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
What each parameter means
The slope is the per-unit rate and the intercept is the starting amount, before any units are used.
Building one
Identify the fixed part and the rate, then write y = rate × x + fixed.
Context restricts the domain
A model of tickets sold makes no sense at x = −3 or x = 2.5. State the values that actually apply.
Careful beyond the data
A model fitted to one range may fail outside it. A growth rate measured over a month rarely holds for a decade.
Answer in words
A modelling answer is a sentence with units, not a bare number. "It costs $47" beats "47".
A rate and a starting value
Most linear situations have a fixed amount plus a per-unit charge — a joining fee plus a monthly rate. Recognising that structure gives you the equation almost immediately.
Say what m and b mean
For a taxi at £3 plus £2 per mile, the slope is the per-mile rate and the intercept the fixed charge. Naming them in context is usually what a modelling question is actually asking for.
The situation restricts the domain
Negative miles and fractional people are mathematically permitted and contextually meaningless. Stating the sensible domain is part of a complete model rather than an optional refinement.
Models have a range of validity
A linear model fitted to a few data points may fail badly outside them. Predicting far beyond the observed range assumes a straight line continues, which nothing in the data supports.
Step 2: Try It Yourself
Tap and try it out.
The slope is 3: for every 1 across, the line goes 3 up.
Step 3: Watch an Example
One step at a time.
Watch Kofi Model a Phone Plan
A plan costs $20 a month plus $0.10 per minute.
- Step 1
The $20 is charged regardless of use, so it is the fixed part.
Step 4: Your Turn
Practice makes it stick.
The Plan
Problem 1 of 2
C = 0.10m + 20. What is the cost for 300 minutes, in dollars?
dollars
The Fee
Problem 2 of 2
A gym charges $30 to join plus $15 a month. Cost after 6 months, in dollars?
dollars
Build the Model
1 of 8
C = 5x + 12. What is the fixed amount?
2 of 8
C = 5x + 12. What is the rate per unit?
3 of 8
C = 5x + 12 at x = 8. What is C?
4 of 8
$25 joining fee plus $10 a month. Cost after 4 months, in dollars?
5 of 8
A tank holds 100 L and drains 4 L a minute. Litres after 10 minutes?
6 of 8
For a model counting tickets sold, is x = 2.5 in the domain? 1 yes, 0 no.
7 of 8
Put the modelling steps in order.
- 1Identify the rate per unit.
- 2Write the equation combining both.
- 3State which input values make sense.
- 4Identify the fixed starting amount.
8 of 8
C = 2x + 40 at x = 15. What is C?
Step 5: Quick Check
Show what you know.
Question 1 of 2
$40 fee plus $6 an hour. Cost for 5 hours, in dollars?
Question 2 of 2
In a cost model, what does the y-intercept represent?
What You Learned
- A linear model is rate times units plus a fixed amount.
- The slope is the per-unit rate; the intercept is the starting value.
- Context restricts which inputs make sense.