How to Write a Linear Equation from a Word Problem

Algebra 1 · Linear functions and slope

How to Write a Linear Equation from a Word Problem

Problem

\text{10 cm tall, grows 3 cm per week}

Answer

y = 3x + 10

Word problems about steady change are linear functions in disguise. The starting value is the y-intercept, and the per-unit change is the slope.

Step-by-step solution

Name the changing amount.
$3\ \text{cm per week}$
Why: A per-week change is the slope, because it repeats for every week.
Name the starting amount.
$10\ \text{cm}$
Why: The height at week 0 is the y-intercept.
Put them in slope-intercept form.
$y = 3x + 10$
Why: The form y = mx + b uses the rate for m and the start for b.

Common mistakes

Swapping the rate and the start: y = 10x + 3.
Writing y = 3x without the starting height.
Using the plant height as the slope because it is the bigger number.

Practice problems

Use the same method. Work each problem on paper, then open the answer to check.

$\text{starts at 20 cm, grows 2 cm per week}$

Show answer

$y = 2x + 20$
$\text{5 L, drains 1 L per minute}$

Show answer

$y = -x + 5$
$\text{40 dollar fee plus 15 dollars per hour}$

Show answer

$y = 15x + 40$
$\text{100 km away, driving 50 km/h closer}$

Show answer

$y = -50x + 100$
$\text{8 cm tall, grows 4 cm per week}$

Show answer

$y = 4x + 8$