Math Core

The more hours you babysit, the more money you earn. The longer a plant grows, the taller it gets. In situations like these, one amount changes because of another. Math has names for the two amounts, and a table, an equation and a graph can all show how they are connected.

Two kinds of variables

Definition

Independent and dependent variables

The independent variable is the amount that changes on its own or that you choose. The dependent variable is the amount that changes because of the independent variable. Its value depends on the other one.

A good test is to put the two amounts into this sentence:

The dependent variable depends on the independent variable.

"The money you earn depends on the hours you work" makes sense. "The hours you work depend on the money you earn" sounds backward. So hours is independent and money is dependent.

Worked example: Name the variables

For each situation, name the independent and dependent variables.

The number of pizzas ordered and the total cost.
The number of minutes a candle burns and its height.

Solutions.

The cost depends on how many pizzas you order. Independent: number of pizzas. Dependent: total cost.
The height depends on how long the candle has burned. Independent: minutes. Dependent: height.

Tables and equations

Maya earns $12 for every hour she babysits. Let $h$ be the hours and $m$ be the money in dollars.

hours, $h$	1	2	3	4	5
money, $m$	12	24	36	48	60

Each money value is 12 times the hours. You can write that rule as an equation:

m = 12h

Writing the equation

Write the equation with the dependent variable alone on one side, like $m = 12h$ . The other side is an expression using the independent variable. Then you can plug in any value of the independent variable to find the dependent one.

With the equation you can go past the table. For 8 hours, $m = 12(8) = 96$ , so Maya earns $96.

Worked example: Find the rule from a table

Write an equation for $y$ in terms of $x$ .

$x$	0	1	2	3	4
$y$	6	7	8	9	10

Compare each pair: $0 \to 6$ , $1 \to 7$ , $2 \to 8$ . Each $y$ is $6$ more than $x$ . The equation is $y = x + 6$ .

Check another pair: $4 + 6 = 10$ . ✓

Graphing the relationship

To graph a relationship, turn each column of the table into an ordered pair $(x, y)$ . Put the independent variable on the horizontal axis and the dependent variable on the vertical axis.

Here are Maya's pairs, $(h, m)$ :

Hours babysitting (across) and money earned in dollars (up).Open in grapher →

The points line up. Every step of 1 hour to the right goes up $12.

Worked example: Use a graph

A bike rider travels at a steady speed. The graph shows time in hours, $t$ , and distance in miles, $d$ .

Time in hours (across) and distance in miles (up).Open in grapher →

Write an equation and find how long it takes to ride 70 miles.

The pairs are $(1, 10)$ , $(2, 20)$ , $(3, 30)$ and so on. Distance is 10 times the time, so $d = 10t$ .

For 70 miles, solve $70 = 10t$ . Divide both sides by $10$ : $t = 7$ . It takes 7 hours.

Common mistake

Don't mix up the axes. The independent variable goes across (the horizontal axis) and the dependent variable goes up (the vertical axis). In an ordered pair, the independent value comes first.

Practice

Practice 1

Jordan mows lawns. The number of lawns he mows affects how much money he makes. Which is the independent variable?

Practice 2

A scientist records the number of days a plant has been growing and the plant's height. Which is the dependent variable?

Practice 3

Pencils cost $4 per pack. The total cost is $c = 4p$ , where $p$ is the number of packs. What is $c$ when $p = 7$ ?

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

Practice 4

The table shows gallons of gas, $g$ , and total cost in dollars, $c$ . Which equation matches?

$g$	1	2	3	4
$c$	3.5	7	10.5	14

Practice 5

The table shows how $y$ depends on $x$ . Write an expression for $y$ in terms of $x$ .

$x$	1	2	3	4
$y$	10	11	12	13

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

Practice 6

The graph shows pairs $(x, y)$ that follow one rule. What is $y$ when $x = 6$ ?

(1, 5)(2, 10)(3, 15)(4, 20)Open in grapher →

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

Practice 7

A cyclist rides at 12 miles per hour, so $d = 12t$ , where $t$ is time in hours and $d$ is distance in miles. How many hours does it take to ride 60 miles?

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

Practice 8

A pool already has 200 gallons of water. A hose adds 25 gallons each minute. The total water is $g = 25m + 200$ , where $m$ is minutes. How many gallons are in the pool after 8 minutes?

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