C. Algebra · Linear and Non-Linear Relations

Graphing ax + by = k and xy = k

C4.2 C4.1 A1About 70 minutes

Learning goals

I can graph a line written in the form ax + by = k using its intercepts.
I can find the slope of ax + by = k by rearranging it.
I can graph xy = k and describe its characteristics: two branches, no intercepts, and inverse variation.
I can find points where these graphs intersect and explain what they mean in a situation.

Words to know

standard form
A linear equation written as ax + by = k, where a, b and k are constants (for example 2x + 3y = 12).

intercept method
Graphing a line by finding where it crosses each axis: set y = 0 to find the x-intercept and x = 0 to find the y-intercept.

hyperbola
The two-part curve that is the graph of xy = k (when k is not 0).

branch
One of the two separate pieces of a hyperbola.

Ticket combinations

The school drama club sells student tickets for $5 and adult tickets for $10. On Friday they took in $100. If s is the number of student tickets and a is the number of adult tickets:

5s + 10a = 100

This is a linear equation in standard form, ax + by = k. Many different combinations work: 20 student tickets and 0 adults, 10 students and 5 adults, 4 students and 8 adults, ... You will learn to graph equations like this quickly, and then meet a very different relation, xy = k.

Graphing ax + by = k: the intercept method

x-intercept: set y = 0 and solve for x.
y-intercept: set x = 0 and solve for y.
Plot both intercepts, check with a third point, and draw the line.

For 2x + 3y = 12: y = 0 gives 2x = 12, so x = 6. x = 0 gives 3y = 12, so y = 4. The line passes through (6, 0) and (0, 4). Check (3, 2): 2(3) + 3(2) = 6 + 6 = 12 ✓.

Slope: rearrange into y = mx + b. 3y = −2x + 12, so y = −(2/3)x + 4. The slope is −2/3. In general, the slope of ax + by = k is −a/b (when b is not 0).

2x + 3y = 12 crosses the axes at (6, 0) and (0, 4). The point (3, 2) is a check: it is on the line because 2(3) + 3(2) = 12.

Worked example

Example 1: A line with a negative coefficient

Graph 3x − 4y = 12. Find its intercepts and slope.

x-intercept: y = 0 gives 3x = 12, so x = 4: (4, 0).
y-intercept: x = 0 gives −4y = 12, so y = −3: (0, −3).
Slope: −4y = −3x + 12. Divide every term by −4: y = (3/4)x − 3. Slope 3/4.
Check (8, 3): 3(8) − 4(3) = 24 − 12 = 12 ✓.

Intercepts (4, 0) and (0, −3); slope 3/4. The line rises to the right.

Worked example

Example 2: The ticket problem

Use 5s + 10a = 100. (a) Find the intercepts and explain them. (b) If 8 adult tickets were sold, how many student tickets were sold? (c) Should the points be joined?

(a) a = 0 gives 5s = 100, s = 20: 20 student tickets and no adults. s = 0 gives 10a = 100, a = 10: 10 adult tickets and no students.
(b) 5s + 10(8) = 100, so 5s = 20 and s = 4.
(c) No. You cannot sell part of a ticket, so only whole-number points on the line make sense: (0, 10), (2, 9), (4, 8), ..., (20, 0).

Intercepts (20, 0) and (0, 10); 8 adults means 4 students; the graph is a set of separate points on the line.

Graphing xy = k

If a trip is 120 km, then speed × time = 120. Going twice as fast takes half the time. Relations like this, where xy = k, are called inverse variation.

Make a table for xy = 6: x = 1, 2, 3, 6 gives y = 6, 3, 2, 1. Negative values work too: x = −1, −2, −3, −6 gives y = −6, −3, −2, −1.

Characteristics of xy = k (k not 0):

The graph is a hyperbola with two branches.
If k > 0, the branches are in quadrants I and III (x and y have the same sign). If k < 0, they are in quadrants II and IV.
It never touches either axis: x = 0 or y = 0 would make the product 0, not k.
As x gets farther from 0, y gets closer to 0, and the other way around.
It is symmetric: if (2, 3) is on xy = 6, then so is (3, 2).

xy = 6 (solid) has branches in quadrants I and III; the dots are (1, 6), (2, 3), (3, 2), (6, 1) and their negatives. xy = −6 (dashed) has branches in quadrants II and IV. Neither graph touches an axis.

	ax + by = k	xy = k
Linear?	Yes	No
Graph	One straight line	A hyperbola with two branches
Intercepts	Usually one x-intercept and one y-intercept	None
Rate of change	Constant: −a/b	Changes; the curve flattens out far from the origin
Example	5s + 10a = 100 (tickets)	speed × time = 120 (a 120 km trip)

Comparing the two kinds of relation.

Worked example

Example 3: Where do x + y = 5 and xy = 4 meet?

Find the points of intersection of x + y = 5 and xy = 4 for positive x. What do they mean if x and y are the side lengths of a rectangle?

Make a table for xy = 4: (0.5, 8), (1, 4), (2, 2), (4, 1), (8, 0.5).
Test which pairs also add to 5: 1 + 4 = 5 ✓ and 4 + 1 = 5 ✓. (2 + 2 = 4, so (2, 2) does not work.)
The graphs cross at (1, 4) and (4, 1).
Meaning: a rectangle with area 4 (xy = 4) and half-perimeter 5 (x + y = 5, so perimeter 10) must be 1 by 4.

(1, 4) and (4, 1). A line and a hyperbola can meet at two points.

The line x + y = 5 (solid) crosses the curve xy = 4 (dashed) at two points, (1, 4) and (4, 1). Each point makes both equations true.

Just for fun

The words hyperbola and hyperbole (an exaggeration, like "I've told you a million times") come from the same Greek word. So the hyperbola is officially the most dramatic curve in math.

Hands-on activity

Coin combinations

You need: about 20 nickels and 10 dimes (or paper slips labelled 5¢ and 10¢), paper, a pencil.

Find as many ways as you can to make exactly $1.00 using only nickels and dimes. Record each way as a point (n, d), where n is the number of nickels and d is the number of dimes.
Write an equation: 5n + 10d = 100.
Plot your points. Do they lie on a straight line? Find the intercepts. Should you join the points?
Now take 24 coins of one kind. Arrange them into equal rows in every way you can (1 row of 24, 2 rows of 12, ...). Record (rows, coins per row) and plot. What shape do the points make?

What to look for: the $1 combinations are (0, 10), (2, 9), (4, 8), ... (20, 0): 11 points on a straight line, not joined. The rows make xy = 24: (1, 24), (2, 12), (3, 8), (4, 6), (6, 4), (8, 3), (12, 2), (24, 1), which curve like a hyperbola branch.

Your turn

Find the intercepts of 3x + 2y = 18.

Find the intercepts of 4x − 5y = 20.

Find the slope of 2x + 3y = 12.

Is (3, 2) on the line 2x + 3y = 12?

Find y on xy = 12 when x = −3 and when x = 0.5.

In which quadrants is the graph of xy = −8?

Tickets: 5s + 10a = 100. How many student tickets were sold if 3 adult tickets were sold?

More practice: warm-up

Find the x-intercept of 2x + y = 8.

Find the y-intercept of 2x + y = 8.

Does the graph of xy = 10 cross the x-axis?

On xy = 20, find y when x = 4.

Which is linear: 7x + 2y = 14 or xy = 14?

More practice: on your own

Find the intercepts, the slope and one more point for 5x + 2y = 20.

Rewrite 6x − 3y = 9 in the form y = mx + b.

A trip is 120 km, so st = 120 (s in km/h, t in hours). Find t for s = 40, 60, 80 and 120. Is the relation linear?

Find the points where x + y = 5 and xy = 6 meet, for positive x.

Nickels and dimes make $2.00: 5n + 10d = 200. Find the intercepts. How many nickels are there if there are 12 dimes?

Challenge

The line 3x + ky = 24 has a y-intercept of 6. Find k and the slope of the line.

A rectangle has an area of 36 cm² and a perimeter of 26 cm. Write two equations and find its dimensions.

Explain why 2x + 3y = 12 and 4x + 6y = 12 never meet, but 2x + 3y = 12 and 4x + 6y = 24 are the same line.

Check your understanding

Quiz

What are the intercepts of 4x + 3y = 24?

What is the slope of x + 2y = 10?

Which point is on xy = 18?

The graph of xy = −5 lies in which quadrants?

Which statement about xy = k (k ≠ 0) is true?

Where do x + y = 7 and xy = 10 meet (positive x)?

For parents and teachers

Big idea: Expectation C4.2 includes graphing ax + by = k and xy = k and identifying their characteristics and the points they define. Standard form is common in real problems (combinations of tickets or coins), and xy = k models inverse variation (speed and time for a fixed distance, length and width for a fixed area).

Common difficulties:

Sign errors when finding intercepts or rearranging, especially with a negative coefficient on y (3x − 4y = 12).
Calling xy = k linear because it has no exponents. Make a table and look at the differences.
Drawing the hyperbola touching an axis. Remind your child that x and y can never be 0.

How to help: The coin activity is quick and concrete. Ask: "Why can't we join these points?" (You cannot have half a coin.)