Real situations that change at a constant rate are modeled by lines. You can build the line from two data points, and you know how to read units. The real skill now is reading what the numbers mean.
Why it works
In $y = mx + b$, the slope $m$ is $\frac{\text{change in } y}{\text{change in } x}$, so its units are “$y$-units per $x$-unit”: dollars per hour, gallons per minute. That makes it a rate. The intercept $b$ is the value of $y$ when $x = 0$, so it is the starting amount: a fee before any work, the water already in a tank. Asking “what are the units?” almost always tells you what a number means.
Key idea
Slope = rate of change, with units of output per input. Intercept = starting value, the output when the input is $0$. Use the model to predict ($x$ given) or to solve ($y$ given).
Worked examples
Worked example: Build and use a model
A plumber charges a \$60 visit fee plus \$40 per hour. Write a model and find the charge for $3.5$ hours.
- Starting value $60$, rate $40$ dollars per hour: $C = 40h + 60$.
- $C = 40(3.5) + 60 = 140 + 60 = 200$.
Answer: $C = 40h + 60$; \$200 for $3.5$ hours.
Worked example: From two data points
A candle is $26$ cm tall after $2$ hours and $18$ cm tall after $6$ hours. Find and interpret the slope and intercept.
- Slope: $\frac{18 - 26}{6 - 2} = \frac{-8}{4} = -2$ cm per hour: it burns $2$ cm each hour.
- Back up $2$ hours from $(2, 26)$: $26 + 2 \cdot 2 = 30$. The candle started $30$ cm tall.
- Model: $H = -2t + 30$. It burns out when $H = 0$: $t = 15$ hours.
Answer: Slope $-2$ cm per hour; intercept $30$ cm.
Common mistake
Reading the $y$-intercept as a rate, or giving a slope without units. In $C = 40h + 60$, the $60$ is a one-time fee (dollars), and the $40$ is dollars per hour. Check by asking what happens when $x$ goes up by $1$: only the slope changes $y$.