Math Core

Tables and graphs show a relationship, but an equation packs the whole relationship into one short line. With an equation, you can find the cost of 7 items or 700 items without making a longer table or a bigger graph.

From "multiply by k" to an equation

In a proportional relationship, you always multiply $x$ by the same number $k$ to get $y$ . Write that sentence in symbols and you have the equation.

Equation of a proportional relationship

Every proportional relationship can be written as

y = kx

where $k$ is the constant of proportionality. And every equation of the form $y = kx$ (with $k$ a number) is a proportional relationship.

For the $3 play tickets from the last lesson, $k = 3$ , so the equation is $y = 3x$ . Test it: 4 tickets give $y = 3 \times 4 = 12$ dollars, which matches the table and the graph.

You can use letters that remind you of the quantities. If $c$ is the cost and $t$ is the number of tickets, write $c = 3t$ .

Writing equations

To write an equation, find $k$ (from a table, a graph or a description), then put it into $y = kx$ .

Worked example: From a description

Apples cost $2.40 per pound. Write an equation for the cost $c$ of $p$ pounds, and use it to find the cost of 3.5 pounds.

The unit rate is 2.40 dollars per pound, so $k = 2.4$ and

c = 2.4p.

When $p = 3.5$ : $c = 2.4 \times 3.5 = 8.4$ . The apples cost $8.40.

Worked example: From a table

Write an equation for this proportional relationship.

$x$	3	6	9
$y$	5	10	15

Find $k$ : $\dfrac{5}{3} = \dfrac{10}{6} = \dfrac{15}{9} = \dfrac{5}{3}$ . So

y = \frac{5}{3}x.

Check with the last column: $\dfrac{5}{3} \times 9 = \dfrac{45}{3} = 15$ . It works.

From a graph, use a point just like in the last lesson. If a line through the origin passes through $(4, 3)$ , then $k = \dfrac{3}{4}$ and the equation is $y = \dfrac{3}{4}x$ .

Recognizing proportional equations

An equation shows a proportional relationship only if it has the form $y = kx$ : a number times $x$ , and nothing added or subtracted.

Worked example: Which equations are proportional?

Decide whether each equation shows a proportional relationship.

$y = 7x$
$y = \dfrac{x}{4}$
$y = x + 4$
$y = 2x - 1$
$y = 0.3x$

Solutions.

Yes, $k = 7$ .
Yes. Dividing by 4 is the same as multiplying by $\dfrac{1}{4}$ , so $y = \dfrac{1}{4}x$ and $k = \dfrac{1}{4}$ .
No. Something is added to $x$ . When $x = 0$ , $y = 4$ , not 0.
No. When $x = 0$ , $y = -1$ . The graph misses the origin.
Yes, $k = 0.3$ .

Common mistake

An equation like $y = 2x - 1$ has a number times $x$ , so it can look proportional. But the extra $-1$ means the graph doesn't go through $(0, 0)$ . Quick check: substitute $x = 0$ . A proportional equation must give $y = 0$ .

Working backward

Sometimes you know $y$ and need $x$ . Put the known value into the equation and ask: " $k$ times what number gives this?" Then divide.

Worked example: How long will it take?

A printer prints pages at a steady rate. The equation $p = 18m$ gives the number of pages $p$ printed in $m$ minutes. How long does it take to print 243 pages?

Substitute $p = 243$ :

243 = 18m

18 times what number is 243? Divide: $m = 243 \div 18 = 13.5$ . It takes 13.5 minutes.

Check: $18 \times 13.5 = 243$ . Correct.

Tip

The value of $k$ in the equation is the same number you see everywhere else: the unit rate in the description, the ratio $\dfrac{y}{x}$ in the table, and the $y$ -value of the point $(1, k)$ on the graph. If these don't agree, look for a mistake.

Practice

Practice 1

Pens cost $6 per box. Which equation gives the cost $c$ in dollars of $b$ boxes?

Practice 2

The equation $y = 4.5x$ shows a proportional relationship. What is $y$ when $x = 8$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 3

This table shows a proportional relationship. Write an equation in the form $y = kx$ .

$x$	2	5	8
$y$	3	7.5	12

Type only the right side, like 2x.

Enter an expression, e.g. 3x^2 - 2x + 1

Practice 4

Which equation does not show a proportional relationship?

Practice 5

A bus travels at a steady speed. The equation $d = 55t$ gives the distance $d$ in miles after $t$ hours. How many hours does it take the bus to travel 220 miles?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 6

A proportional relationship has a graph that passes through $(4, 3)$ . Write its equation in the form $y = kx$ .

Type only the right side, like 2x.

y = 0.75x(4, 3)Open in grapher →

Enter an expression, e.g. 3x^2 - 2x + 1

Practice 7

A muffin recipe uses 3 cups of flour for every 2 cups of sugar. The equation $f = ks$ gives the cups of flour $f$ for $s$ cups of sugar. How many cups of flour are needed for 5 cups of sugar?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 8

Faucet A fills a tub according to the equation $w = 2.5t$ , where $w$ is gallons of water and $t$ is minutes. Faucet B fills 18 gallons in 6 minutes at a steady rate. In 20 minutes, how many more gallons does Faucet B fill than Faucet A?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.