Algebra
Data and statistics
Interpreting slope and intercept of a linear model
20 practice questions
2 video lessons
Theory + worked examples
Theory
In a linear model \(y=mx+b\):
- the slope \(m\) is the rate of change (\(y\)-units per \(x\)-unit),
- the intercept \(b\) is the starting value (at \(x=0\)).
Attach real units — a slope is “so much per something.”
Slope as a rate, intercept as a start.
Interpreting a linear model.
Meanings:
\[m=\text{rate of change},\qquad b=\text{starting value}\]
Context gives the units for each.
How to interpret a model
- Identify the slope and intercept.
- Read the slope as a rate with units.
- Read the intercept as the value at \(x=0\).
- State each meaning in context.
Example 1 — Interpret the slope
For cost \(C=15h+40\) (dollars, hours), interpret \(15\).
Solution
The slope is a rate.
| \(15\) | \(=\) | \(\$15\text{ per hour}\) |
Example 2 — Interpret the intercept
For \(C=15h+40\), interpret \(40\).
Solution
The intercept is the value at \(h=0\).
| \(40\) | \(=\) | \(\$40\text{ fixed fee}\) |
Example 3 — Predict
Find the cost for \(3\) hours using \(C=15h+40\).
Solution
Substitute \(h=3\).
| \(C\) | \(=\) | \(15(3)+40=\$85\) |
Example 4 — Units matter
Why attach units to the slope?
Solution
The slope is a rate — \(y\)-units per \(x\)-unit — so units give it meaning.
Common pitfalls
The slope is a rate, not just a number.
The intercept is the value at \(x=0\).
Always attach units in context.
Frequently asked questions
What does the slope of a model mean?
The rate of change — how \(y\) changes per unit \(x\).
What does the y-intercept mean?
The starting value at \(x=0\).
In \(C=15h+40\), what is \(15\)?
\(\$15\) per hour.
Why attach units?
They give the slope and intercept real meaning.
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