Linear Modeling on the SAT
Build the Equation from Context
Linear modeling is how you turn a story into an equation. When a taxi charges a flat fee plus a per mile rate, or a savings account grows by the same amount each month, that is a linear model.
The Definition
Slope ($m$)
The rate of change or change per unit.
Y-Intercept ($b$)
The starting value when $x = 0$.
Step-by-Step Guide
A table includes points $(1, 14)$ and $(2, 18)$. What linear equation models the data?
Find the Slope ($m$)
Compute the change in $y$ over the change in $x$. Look for the difference between the second point's value and the first.
Slope found: 4
Solve for the Intercept ($b$)
Use the slope-intercept form $y = mx + b$. Plug in the slope you just found ($m = 4$) and one known point, for example $(1, 14)$.
Intercept found: 10
Write the Final Model
Combine the slope and intercept into the final equation.
Desmos Regression
Desmos is faster when you have many data points or need to verify quickly. Enter your data into a table, then type the regression formula:
Practice Questions
Translate each context into a linear equation.
A gym charges $25$ to join and $15$ per month. Which equation models total cost $C$ after $m$ months?
A candle is $20$ cm tall and burns $3$ cm per hour. What is the equation for height $h$ after $t$ hours?
Two points on a line are $(2,7)$ and $(6,19)$. Which equation models the line?
A line models a savings account that increases by $50$ dollars every week and starts at $200$. What is the value after $8$ weeks?
Key Takeaways
Slope is the rate.
Intercept is the starting value.
Match the equation parts to the story.
About the Instructor
