Linear equation word problems

How the SAT tests this, with worked examples — plus the 4 skills it covers. Free to read, no account needed.

These questions describe a situation in words and ask you to model it, almost always with

y = mx + b

. The difficulty is translation, not algebra.

A small vocabulary maps phrases onto parts of the equation reliably.

A fixed amount that exists before anything happens is the intercept

b

: a joining fee, a starting balance, an initial temperature. An amount that repeats per unit is the rate

m

: per hour, per mile, each month.

So "a gym charges a

\$40

joining fee plus

\$25

a month" becomes

y = 25x + 40

, where

x

is months and

y

is total cost. The reliable test is asking which number happens once and which repeats.

Watch for questions asking what a value means in context rather than asking you to solve. In

y = 25x + 40

, the

25

is the monthly charge and the

40

the one-off fee — and the answer choices will offer those two swapped.

Worked examples

A plumber charges a

\$60

call-out fee plus

\$45

per hour. Write the cost

C

for

h

hours.
The call-out happens once, so it is the intercept; the hourly rate repeats

{C = 45h + 60}

A tank holds

500

litres and drains at

12

litres per minute. Write the volume

V

after

t

minutes.
The starting amount is the intercept, and the rate is negative because the volume falls

{V = 500 - 12t}

For

C = 45h + 60

, what does the

60

represent?

It is the value of

C

when

h = 0

, so it is the charge before any hours are worked, making (B) correct. (A) describes the

45

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