Linear relationship word problems

How the SAT tests this, with worked examples — plus the 3 skills it covers. Free to read, no account needed.

A linear relationship is one where a quantity changes by the same amount each step. The model is y=mx+by = mx + b, and the work is deciding which number in the words is the rate and which is the starting value.
et_word_problem_translation.png
The number that happens once, before anything else, is the intercept bb. The number that repeats per unit is the rate mm. A fee of $40\$40 plus $25\$25 a month gives y=25x+40y = 25x + 40.
When a question gives you two data points instead of a rate, find the rate yourself
m=change in ychange in x{m = \frac{\text{change in } y}{\text{change in } x}}
then substitute one point back to get bb.
A decreasing quantity simply has a negative rate. A tank draining at 1212 litres a minute from 500500 litres is V=500−12tV = 500 - 12t.
Many questions never ask you to solve. They ask what a number means in context, and the answer choices offer the rate and the starting value swapped — so always check which one happens repeatedly.

Worked examples

A membership costs $30\$30 to join and $12\$12 a month. Write the total cost CC after mm months.
The joining fee happens once; the monthly charge repeats
C=12m+30{C = 12m + 30}
A plant is 88 cm tall after 22 weeks and 2020 cm after 55 weeks. Find the weekly growth rate.
Use the change in each quantity
m=20−85−2=4 cm per week{m = \frac{20 - 8}{5 - 2} = 4 \text{ cm per week}}
For C=12m+30C = 12m + 30, what does 3030 mean?
  • AThe monthly charge
  • BThe cost before any months have passed
It is CC when m=0m = 0, so (B) is correct. (A) describes 1212.

Is linear relationship word problems costing you points?

Take the free 10-question diagnostic for a predicted SAT score and a breakdown of which domains are costing you points.

Predict your SAT score →

All the skills in Linear relationship word problems