Writing Linear Models from Context
Many SAT questions describe a situation in words and ask you to build the equation. The trick is separating the one-time amount from the per-unit rate.
Reading the concept isn't enough. The score comes from practicing the real Bluebook-style format and finding the exact mistakes you keep making.
What the SAT tests here
A fixed starting amount is the constant (the intercept); an amount that repeats per unit is the coefficient (the slope). Decide which quantity depends on the other before you write anything.
How to solve one, step by step
Example: a gym charges a $\$30$ joining fee plus $\$5$ per visit.
- The $\$5$ repeats per visit — that's the rate (slope).
- The $\$30$ happens once — that's the starting value.
- Cost for $v$ visits: $C = 5v + 30$.
The mistakes that cost points
- Reversing the variables. Be clear about which quantity is the input and which is the output.
- Treating a fixed fee as a rate. A one-time charge is a constant, not multiplied by the count.
Practice questions
Try these the way you would on test day, then open the solution to check your method.
Easy
A plumber charges a one-time service fee of $\$40$ and $\$25$ per hour of work. A customer's bill for $t$ hours is $\$B$. Which equation correctly represents the bill?
- A$B = 40t + 25$
- B$B = 25t$
- C$t = 25B + 40$
- D$B = 25t + 40$
Show solution
Answer: D, $B = 25t + 40$. The $\$40$ service fee is fixed regardless of time, so it is the constant term. The $\$25$ per hour is the rate, giving $25t$. Therefore $B = 25t + 40$.
Medium
A caterer charges $\$15$ per guest and requires a non-refundable deposit of $\$200$ when the contract is signed. The total charge for $g$ guests is $\$C$. A contract also states that the guest count cannot fall below 10. Which equation correctly models the total catering charge?
- A$C = 200g + 15$
- B$C = 15g + 200$
- C$g = 15C + 200$
- D$C = 15g$
Show solution
Answer: B, $C = 15g + 200$. The $\$200$ deposit is a fixed fee paid once — it becomes the constant term. The per-guest charge of $\$15$ multiplied by $g$ gives $15g$. So $C = 15g + 200$. The minimum guest requirement of 10 is a domain constraint, not part of the cost equation.
Hard
A streaming service charges a monthly subscription fee of $\$12$ and an additional $\$4$ per premium movie rented. A customer rents $m$ premium movies in a month and is charged a total of $\$T$. The service also applies a $\$2$ late-return fee for each movie returned late; the customer returned $k$ movies late that month. Which equation models the customer's total charge?
- A$m = 4T + 2k + 12$
- B$T = 4m + 12$
- C$T = 4m + 2k + 12$
- D$T = 12m + 4k + 2$
Show solution
Answer: C, $T = 4m + 2k + 12$. The monthly subscription of $\$12$ is a fixed cost. The per-movie rental charge is $\$4 \times m$. The late fee is $\$2 \times k$. Adding all three: $T = 4m + 2k + 12$.
Find your exact gaps
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