Digital SAT · Linear Functions
How to Solve Linear Word Problems on the Digital SAT
A steady-rate situation becomes y = mx + b. Two numbers do all the work: the amount that repeats per unit is the slope, and the fixed amount you start with is the intercept. Read which is which, hang the repeating one on the variable, and add the fixed one. When the choices are equations, drop a data point from the problem into each and keep the one that holds true.
- per test
- 2 per test per test
- typical difficulty
- Easy to medium typical difficulty
- practice questions
- 146 practice questions
Frequency reflects how often this question type appears on a full-length Digital SAT. The difficulty mix reflects every question of this type across our bank.
What the question bank shows
The per-unit amount is the slope; the one-time amount is the intercept. Placing them is the whole task.
Drop a given point into each candidate equation and keep the one that holds.
Most offer competing equations, so a single data point clears the field.
How to recognize linear word problems questions
- Something changes by the same amount per unit: dollars per month, centimeters per hour, cost per item.
- There is a one-time starting amount plus a steady rate, like a joining fee plus a monthly charge.
- The question asks which equation fits, or for a value the model predicts.
- Choices are equations in y = mx + b form that differ in slope or intercept.
Why students miss these
The step-by-step method
- 1
Define the variables
Say what x and y stand for, with units, so slope and intercept have clear meaning.
- 2
Find the per-unit rate
The amount per unit is the slope m; if the quantity drops as x grows, m is negative.
- 3
Find the starting amount
The value when x is zero is the intercept b, such as a joining fee or an initial height.
- 4
Assemble and test
Write y = mx + b, then check it against a data point from the problem.
Solving it on Desmos
- Graph your equation. Type the equation you built as y = ... to see its line.
- Pass it through a known point. Confirm the line hits a point the problem gives, like the cost at a stated number of units.
- Eliminate the rest. Graph each candidate and keep the line that fits the starting value and the data point.
Full Desmos walkthrough for linear word problems→
Worked examples
Easy example
A candle is centimeters tall and burns down at a constant rate of centimeters per hour. The candle has been burning for hours.
A: Correct. The candle starts at 18 cm and loses 2 cm each hour, so .
B: Incorrect. This adds height each hour, but the candle burns down, so the rate is negative.
C: Incorrect. This swaps the starting height and the rate; 18 is the starting height, not the rate.
D: Incorrect. This swaps the values and uses the wrong sign on the starting height.
Explanation
The starting height 18 is the constant and the burn rate is per hour, giving .
Medium example
A: Correct. The joining fee 30 is the constant and 22 is the monthly rate, so .
B: Incorrect. This swaps the joining fee and the monthly fee.
C: Incorrect. This drops the one-time joining fee.
D: Incorrect. Monthly fees add to the total, so the rate should be positive.
Explanation
Joining fee 30 plus 22 per month gives T = 30 + 22m.
Hard example
A: Incorrect. This swaps the intercept and the rate.
B: Incorrect. 90 is the cost at 4 units, not the intercept at 0 units.
C: Incorrect. The rate is 10, not 15.
D: Correct. The rate is ; the intercept is , so .
Explanation
Rate 10, intercept 50: C = 50 + 10n.
The common traps
| Pattern | What it does | The tell |
|---|---|---|
| Rate and start swapped | Hung the one-time amount on the variable, or used the per-unit rate as the intercept. | At x = 0 the model must equal the fixed starting amount, not the rate. |
| Missed a negative rate | Used a positive slope for a quantity that decreases. | If the amount falls as time grows, the rate m is negative. |
| Followed word order | Placed numbers as they appear in the sentence, not by their role. | Decide which number is per-unit and which is one-time before writing. |
| Answered a value, not the model | Computed one output when the equation was asked, or the reverse. | Check whether the question wants the model or a value it predicts. |
Try it: two real questions
The graph shows the number of candy bars a machine has wrapped after a given number of seconds. At what rate does the quantity change each second?
The line in the graph represents the number of liters of water in a draining tank after a given number of minutes. By how many units does the quantity change per minute?
Question 3 is ready when you are
Keep going with more questions of this type, each with the same step-by-step reasoning. Free to start.
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Related reading
Common questions
How do you build a linear equation from a word problem on the SAT?
Write y = mx + b. Set m to the per-unit rate and b to the value when x is zero. Hang the rate on the variable, add the fixed amount, and test the equation against a data point.
How do I tell the slope from the y-intercept in a word problem?
The slope is the amount that repeats per unit, like a monthly fee. The intercept is the one-time or starting amount that does not repeat, like a joining fee or an initial height.
What if the quantity decreases over time?
Then the slope is negative. A candle burning down or a balance being paid off drops as time grows, so the rate is negative even when the numbers in the problem are positive.
Can Desmos check a linear model for me?
Yes. Graph the equation and confirm its line passes through a point the problem gives. If several equations are offered, graph each and keep the one that fits the start and the data point.
Why do I keep picking the wrong equation on these?
Usually the rate and the starting value are swapped, or a decrease was written with a positive slope. Decide which number is per-unit and which is one-time first, then test at x = 0.
Practice linear word problems the way it is tested
Start with the free 16-question diagnostic, then drill this type with step-by-step reasoning on every question.
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