How to go from a paragraph of text to a clean equation — and what the SAT is actually testing when it gives you these.
Here's the thing about linear equation word problems: the math is usually not the hard part. If someone just handed you y = 3x + 50, you'd know exactly what to do with it. The challenge is getting to that equation when it's hidden inside a paragraph about gym memberships or delivery fees.
That's what this article is about — building the habit of reading a word problem and knowing exactly how to pull a linear equation out of it.
How to know it's a linear problem
Not every word problem is linear, so the first thing you want to do is figure out whether you're dealing with a linear relationship. The pattern is almost always the same: there's a fixed starting amount and a constant rate of change.
Think about it like this. If a gym charges a $40 sign-up fee and then $25 per month, the total cost grows by the same amount every month. That's linear. If the cost doubled every month instead, that wouldn't be linear — that's exponential growth, and a different topic entirely.
Look for language that suggests something is happening at a steady, unchanging rate. Words like “per,” “each,” “every,” or “for each additional” are usually signals that you're dealing with a linear relationship.
Translating words into math
Once you know it's linear, the next step is pulling out the pieces you need. Every linear equation has two core parts:
Slope (rate of change)
The amount that changes for each unit increase. This is the “per month,” “per hour,” or “for every additional item” part of the problem.
That number becomes m.
Y-intercept (starting value)
The amount that exists before any change happens. This is the “initial fee,” “base price,” “already has,” or “starts with” part of the problem.
That value becomes b.
The SAT loves to use specific words that map directly to these two parts. Here's a quick reference:
| Word / phrase | What it usually represents |
|---|---|
| “per,” “each,” “every,” “for each additional” | Slope (rate of change) |
| “initial,” “starting,” “base,” “flat fee,” “already” | Y-intercept (starting value) |
| “total,” “combined,” “overall cost” | The output — your y |
| “number of,” “how many” | The input — your x |
Once you've identified these parts, you plug them straight into slope-intercept form:
where m is the slope and b is the y-intercept.
Interpreting slope and y-intercept in context
This is where a lot of students lose points — not because the math is hard, but because the SAT asks you to explain what the numbers mean.
You'll see questions like: “In the equation C = 0.15m + 35, what does the 35 represent?” The answer isn't “the y-intercept.” That's the math term, but the SAT wants you to connect it back to the situation. In this case, it might be a base monthly charge before any usage is added.
The slope always tells you how much something changes per unit. The y-intercept always tells you the value when the input is zero — meaning before anything has happened yet.
For the slope, always ask yourself: “for every additional what, how much does what change?” That framing will almost always give you the correct interpretation. If the equation is C = 0.15m + 35 and m is miles driven, then 0.15 means the cost increases by $0.15 for every additional mile driven.
Worked examples
10x + 15y = 85 represents this situation, where x is the number of on-site training courses and y is the number of online training courses this apprentice has enrolled in. How many more hours does each online training course take than each on-site training course?Quick test
Try these on your own. Pick an answer and you'll see right away whether it's correct, along with the reasoning.
c is the total number of calories the veterinarian recommends the rabbit should eat each day if the rabbit's weight is x pounds?Q for a certain product as a linear function of the selling price P. The demand was 20,000 units when the selling price was $40 per unit, and the demand was 15,000 units when the selling price was $60 per unit. Based on the model, what is the demand, in units, when the selling price is $55 per unit?