A picture of a relationship
45 min
Two hosts talk the lesson through. The voices are synthetic; the script was written from this lesson and checked against it, and asserts nothing the lesson does not.
- Read a straight line for its starting value and its rate, and say what each one means in the situation
- Write the equation of a line from a described situation, and build its table
- Explain what makes a relationship straight, and what a slope means in the situation's own units
Lesson 1 wrote the phone plan as 12 + 0.05m. Lesson 3 solved equations like it. This lesson draws it, and the drawing turns out to contain both of the numbers you started with, in plain sight.
Three ways of saying the same thing
Take the taxi from lesson 1's practice: $3.50 to get in, $1.20 a mile. That relationship can be written three ways.
As a sentence. The fare starts at three dollars fifty and grows by a dollar twenty for every mile.
As a table.
Miles (d) |
Fare |
|---|---|
| 0 | $3.50 |
| 2 | $5.90 |
| 5 | $9.50 |
| 10 | $15.50 |
As an equation. fare = 3.50 + 1.20d
These aren't three topics. They are one relationship in three costumes, and being able to move between them is most of what this lesson is for. The table came from the equation by substituting, which is lesson 1. The graph below comes from the table by plotting.
What the two numbers are
Look at the equation and the picture together, because each number in one is a feature of the other.
The 3.50 is where the line starts. It is the fare at zero miles, which is the flag charge you pay for getting in. On the graph it's the height of the line where it meets the vertical axis. Its usual name is the intercept.
The 1.20 is the rate. It is how much the fare goes up for each extra mile, and on the graph it's how steeply the line climbs. Its usual name is the slope or gradient.
That is the whole of it. Any relationship of the form y = starting value + rate × x is a straight line, and any straight line can be written that way. The general form you'll meet everywhere is:
y = mx + c
where m is the rate and c is the starting value. The letters are a convention, not a law; some countries write y = mx + b. What matters is which number does which job.
The phone plan from lesson 1 was 12 + 0.05m. What is its starting value, what is its rate, and what does each one mean in the situation?
Show the answer
The starting value is 12 and it is the monthly charge you pay before making a single call, so it's the height of the line at zero minutes. The rate is 0.05, which is five cents per minute, and it is how steeply the cost climbs as you talk. Notice that the rate is small, so the line is nearly flat, which is a true fact about the plan: minutes are cheap and the fixed charge dominates until you have talked for a long time. At two hundred and forty minutes the calls have cost twelve dollars, and only then does the variable part match the fixed part.
Reading a slope in the situation's own units
The slope of a line is a number, and on its own it means very little. What makes it useful is that it always has units, and the units come from the situation.
For the taxi, the slope is 1.20 dollars per mile. For the phone plan it's 0.05 dollars per minute. If you plotted the distance a car travels against time, the slope would be miles per hour, which is a thing you already know how to think about: speed is a slope.
To find a slope from any two points on a line, take the change in the up-and-down direction and divide by the change in the across direction. The two marked points on the graph are at two miles, where the fare is $5.90, and at five miles, where it is $9.50. Work the slope out from those two points before you open the answer.
What slope do the two marked points give?
Show the answer
Change in fare divided by change in miles: (9.50 - 5.90) / (5 - 2) = 3.60 / 3 = 1.20 dollars per mile. Which is the number that was in the equation all along.
Any two points on the line give the same answer, and that's what makes the line straight: the rate is the same everywhere on it. A relationship where the rate changes as you go is not a straight line, and this course doesn't cover those.
A savings account starts with $20 in it and $5 is added every week, with no interest. Write the equation, and say what the graph of it looks like at week zero and how steeply it climbs.
Show the answer
total = 20 + 5w, where w is the number of weeks. At week zero the line is at a height of 20, since that is the money already there. It climbs by 5 dollars per week, so it's five times steeper than a line rising a dollar a week. Notice how little of this is about algebra: the equation is a way of writing down something you could have said in a sentence, and the graph is a way of seeing it.
Building a line from a situation
The situation. A printer costs $180 to buy, and each page costs 4 cents to print.
The equation. The cost that does not depend on pages is 180, and the rate is 0.04 per page. So cost = 180 + 0.04p.
The table, by substituting, which is lesson 1's skill doing a job here.
Pages (p) |
Cost |
|---|---|
| 0 | $180 |
| 500 | $200 |
| 2,000 | $260 |
| 5,000 | $380 |
The question the picture answers instantly. Someone offers you a different printer at $40, with pages at 8 cents. Which is cheaper? It depends on how much you print, and the two lines cross at the point where the answer changes. Setting the two costs equal and solving is lesson 3's method:
180 + 0.04p = 40 + 0.08p
Take 0.04p off both sides: 180 = 40 + 0.04p. Take 40 off both: 140 = 0.04p. Divide by 0.04: p = 3,500.
So at three and a half thousand pages the two printers cost the same, and below that the cheap printer wins while above it the expensive one does. That's what solving an equation means when you have a picture: finding where two lines cross.
Check that answer before trusting it. What does each printer cost at exactly 3,500 pages?
Show the answer
The first is 180 + 0.04 × 3500, which is 180 + 140, so $320. The second is 40 + 0.08 × 3500, which is 40 + 280, so $320 as well. The two agree, so 3,500 is right. This is the same substitution check from lesson 3, and it is worth noticing that it works identically here even though the question arrived as a shopping decision rather than as an equation.
A second taxi firm charges $2.80 to get in and $1.40 a mile. Here is its table, and the equation worked to the last line. The fixed part is 2.80. The rate is 1.40 per mile. So the fare is 2.80 + something. Finish the equation, then use it to find the fare for five miles.
Show the answer
| Miles | Fare |
|---|---|
| 0 | $2.80 |
| 2 | $5.60 |
| 10 | $16.80 |
fare = 2.80 + 1.40d, and at five miles that's 2.80 + 7.00, which is $9.80. Notice that this firm is cheaper than the first one for short trips, because its flag charge is lower, and dearer for long ones, because its rate is higher. The two lines cross somewhere, and finding where is exactly the printer question above.
What people get wrong
Reading a slope without its units. A slope of 1.20 means nothing until you say dollars per mile. Steeper does not mean bigger unless the two lines measure the same things.
Thinking the intercept is always the answer to something sensible. In the taxi it's a real charge. In a line fitted to data, the value at zero can be meaningless or impossible, and you should ask before using it.
Confusing the rate with the total. The taxi's slope is 1.20, and the fare after five miles is 9.50. The slope is not the fare; it's how fast the fare grows.
Expecting every relationship to be straight. Straight means a constant rate. Compound growth, the shape of the savings curve you may meet elsewhere, is not straight, because its rate depends on the current amount.
- A gym charges $30 to join and $22 a month. Write the equation for the total cost after
mmonths, and give the cost at 6 months. - What is the starting value and what is the rate in
y = 8 + 3x? - A line passes through the points (0, 20) and (5, 45). What is its slope, and what is its equation?
- A phone plan is
y = 15 + 0.02m. How many minutes make the bill $25? - Two plans: plan A is
20 + 0.10m, plan B is35 + 0.04m. At how many minutes do they cost the same? - A slope is 4. Say what it means if the axes are hours and dollars, and what it means if they are miles and litres.
- Sketch
y = 6 + 2xforxfrom 0 to 5. Mark the starting value on your sketch, and mark one step of the slope.
Answers to the practice set, once you have done all seven
Show the answer
30 + 22m, and at six months30 + 132, which is $162.Starting value 8, rate 3.
Slope is
(45 - 20) / 5, which is 5, and the equation isy = 20 + 5x.Take 15 off both sides to get
10 = 0.02m, then divide, so 500 minutes. Check:15 + 0.02 × 500is 25.Set them equal:
20 + 0.10m = 35 + 0.04m, so0.06m = 15, som = 250. Check: A gives20 + 25, which is 45, and B gives35 + 10, which is 45.Four dollars per hour in the first case, and four litres per mile in the second, which would be an extraordinarily thirsty vehicle. The number is the same and the meaning comes entirely from the units.
It starts at a height of 6 and climbs 2 for every 1 across, so it passes through (0, 6), (1, 8) and (5, 16).
Connections
Lesson 1 built expressions and evaluated them, which is exactly how a table is made.
Lesson 3's method for solving turns up here as finding where two lines cross, and the printer example is that idea in its natural habitat.
The lesson after this turns sentences into equations, which is the step that comes before everything on this page: somebody has to write 180 + 0.04p before anybody can draw it.
The last lesson handles two equations at once, and the picture of that is two lines and the point they share.
Go deeper
- OpenStax, Elementary Algebra 2e, chapter 4, free online, on graphing lines, slope and intercepts, with more worked examples.
- Khan Academy, linear equations and graphs, for practice reading and drawing lines.
Check your understanding
This lesson has a 6-question quiz. Pass it and the questions come back on a schedule in Review, so what you learned stays learned. Your progress is saved in your browser; no account needed.