Part 12 of 20 · LinesSee the whole course
How to solve linear equation word problems
The rate (per visit, per kilometer, per hour) is the slope m. The starting amount (a joining fee, a base fare) is b. Write y = mx + b, then put a number in for x, or solve for x when you know the total.
Turn the story into y = mx + b
Every linear word problem hides the same two numbers. Find them and the equation writes itself:
- Say what x and y are. x is what you count (visits, kilometers), y is the total that results (the cost).
- Find the rate. Words like “per”, “each” and “every” point at it. This is m, the slope.
- Find the start. It is paid or counted once, before anything happens. This is b.
- Write y = mx + b. Then read the question: do you know x, or do you know y?
That last step is the whole game. If you are given x, you put it in. If you are given the total y, you solve for x, exactly as in solving a linear equation.
Example 1: the gym
A gym costs 20 to join and 5 per visit. The 5 is paid every time, so m = 5. The 20 is paid once, so b = 20.
y = 5x + 20
How much after 6 visits? Put 6 in for x.
y = 5(6) + 20 = 30 + 20 = 50
How many visits for 70? Put 70 in for y and solve.
5x + 20 = 70
5x = 50
x = 10 visits
Example 2: the taxi
A taxi charges a fare of 3, plus 2 for every kilometer. So m = 2 and b = 3, and y = 2x + 3.
A 7 km ride: y = 2(7) + 3 = 17
25 in your pocket: 2x + 3 = 25, so 2x = 22 and x = 11 km
Try both stories below. Switch between “Find the cost” and “Work backwards” and watch the same line answer both questions.
Check the answer in the story
A word problem is not finished until the answer makes sense. Ten visits cost 5(10) + 20 = 70, which is exactly the money you had. An answer like 10.4 visits, or a negative number of kilometers, tells you to look again.
When the rate goes down
Not every story climbs. A candle that is 20 cm tall and burns 2 cm every hour has a negative slope: y = −2x + 20. Here b is the height you start at, and the candle is gone when y = 0.
Watch it explained in under a minute
The five mistakes to avoid
- Swapping the fee and the rate. The fee is paid once (b). The rate is paid each time (m).
- Using a total as the rate. If 6 visits and a fee came to 50, the rate is not 50. Take away the fee first: (50 − 20) ÷ 6 = 5.
- Answering the wrong way round. Check what the question asks for: the cost (y) or the number of visits (x).
- Skipping the check. Put your answer back into the story to see that it works.
- Leaving out the unit. Finish with a sentence: “It takes 10 visits.”
Try it yourself
Write the equation first, then answer. Tap a question to see the answer.
1. A pool costs 12 to join and 3 per swim. What is the cost after 8 swims, and how many swims can you afford with 30?
y = 3x + 12. After 8 swims: 3(8) + 12 = 36. For 30: 3x + 12 = 30, so 3x = 18 and x = 6 swims.
2. A candle is 20 cm tall and burns 2 cm every hour. How tall is it after 5 hours, and when does it burn out?
y = −2x + 20. After 5 hours: −2(5) + 20 = 10 cm. It is gone when y = 0: 0 = −2x + 20, so x = 10 hours.
3. A plant is 10 cm tall and grows 3 cm every week. When is it 25 cm tall?
y = 3x + 10. Set y = 25: 3x + 10 = 25, so 3x = 15 and x = 5 weeks.
Where word problems fit in
A story gives you the slope and b in plain words, which is why slope-intercept form is so useful. Next in this course: parallel and perpendicular lines.