Chapter 10 · Relationships between two variables

Independent and Dependent Variables: Rules from Tables

Name the independent and dependent variables in a situation, complete a table of their values, and write the equation that fits every row.

17 min read

Markers come in packs of 8. Buy 1 pack and you have 8 markers. Buy 3 packs and you have $8 \times 3 = 24$ markers. When the number of packs changes, the number of markers changes with it.

A link like this between two quantities is called a relationship. In this lesson you will name the roles of the two quantities, fill in a table of their values, and write the relationship as an equation.

A variable is a letter that stands for a number. Here each variable stands for a quantity that can take different values. Let $p$ be the number of packs you buy. Let $m$ be the total number of markers you get.

A table can list values of $p$ and $m$ that go together. In the table below, the first column counts packs and the second column counts markers. Each row is one pair: a number of packs and the number of markers it gives.

Marker table: p is the number of packs, m is the total number of markers

Number of packs, p	Total markers, m
$1$	$8$
$2$	$16$
$3$	$24$
$4$	$32$

When one quantity depends on another, the two variables have different roles.

The independent variable stands for the quantity whose value comes first. You choose that value, or the situation gives it to you. You choose how many packs to buy, so $p$ is the independent variable.
The dependent variable stands for the quantity whose value depends on the first one. You work it out from the independent value. The number of markers depends on the number of packs, so $m$ is the dependent variable.

The situation decides the roles. The letters do not decide them, and neither does the order in which the quantities are named. Ask two questions: Which value comes first? Which value depends on it?

In every table in this lesson, the independent variable is in the first column and the dependent variable is in the second column.

Worked example

Naming the two roles

One lap of a pool is 50 meters long. Let $d$ be the total distance you swim in meters. Let $n$ be the number of laps you swim. You decide how many laps to swim. Which variable is independent, and which is dependent?

List the quantities that can change. The number of laps can change, and so can the total distance. The 50 meters in one lap stays the same, so it is not a variable.
Ask which value comes first. You decide how many laps to swim, so the number of laps comes first. The variable for the laps is the independent variable.
$\text{independent variable: } n$
Ask which value depends on it. Once you know the laps, you can work out the distance. For example, 3 laps give 150 meters. So the distance depends on the laps.
$50 \times 3 = 150$
Name the dependent variable, and check that the order of naming did not fool you. The problem names the distance first, but the distance still depends on the laps. The roles come from the situation.
$\text{dependent variable: } d$

Answer

The number of laps, $n$ , is the independent variable. The total distance in meters, $d$ , is the dependent variable, because the distance depends on the number of laps.

Key concept

One rule links the two variables

When one quantity depends on another, the independent variable stands for the quantity whose value comes first. The dependent variable stands for the quantity whose value depends on it.

A rule tells you how to work out each dependent value from its independent value. A table shows the rule at work: each column counts one quantity, and each row is one pair of values that go together. An equation states the same rule once, with variables, so that it fits every row. For the marker packs, the rule is multiply the number of packs by 8, and the equation is $m = 8p$ .

To complete a table, you find the values that are missing. A table for one relationship uses one rule, so every row must fit it. In this lesson, each rule either multiplies by the same number every time (never 0) or adds the same number every time.

First say what each column counts and what one row stands for. The independent value is the input that goes into the rule. The dependent value is the output that comes out.

If the dependent value is missing, apply the rule to the independent value in that row.
If the independent value is missing, the roles stay the same, so work backwards. Write the rule for that row with the value you know. This gives a one-step equation. Solve it with the inverse operation.

The next table comes from a bottling machine. Three of its values are missing, and a question mark shows each one.

Bottle table (to complete): t is the number of minutes the machine runs, b is the number of bottles filled

Minutes the machine runs, t	Bottles filled, b
$2$	$18$
$5$	?
$8$	?
?	$108$

Worked example

Completing a table

A machine fills 9 bottles every minute. Let $t$ be the number of minutes the machine runs. Let $b$ be the number of bottles it fills. Find the three missing values in the bottle table above.

Say what the table counts and name the roles. The first column counts minutes and the second counts bottles. Each row is one run of the machine. You choose how long the machine runs, and the number of bottles depends on that time.
$\text{independent: } t \qquad \text{dependent: } b$
State the rule in words and test it on the complete row. The machine fills 9 bottles each minute, so the number of bottles is 9 times the number of minutes. The first row agrees: 2 minutes give 18 bottles.
$9 \times 2 = 18$
In the second and third rows, the dependent value is missing. The rule gives the bottles from the minutes, so multiply the minutes in each row by 9.
$9 \times 5 = 45, \quad 9 \times 8 = 72$
In the last row, the independent value is missing. The roles stay the same, because the bottles still depend on the minutes. So work backwards: 9 times the minutes equals the bottles, and here the bottles are 108. Write that as a one-step equation for this row.
$9t = 108$
Divide both sides by 9. Dividing by 9 undoes multiplying by 9, so only t is left on the left side. The two sides were equal, and you divide both by the same non-zero number, so they stay equal.
$9t \div 9 = 108 \div 9, \quad \text{so } t = 12$
Check the last row with the rule. 12 minutes should give 108 bottles, and they do.
$9 \times 12 = 108$

Answer

The missing values are $45$ bottles for $5$ minutes, $72$ bottles for $8$ minutes, and $12$ minutes for $108$ bottles.

Bottle table (completed)

Minutes the machine runs, t	Bottles filled, b
$2$	$18$
$5$	$45$
$8$	$72$
$12$	$108$

Look again at the completed bottle table. In every row, the number of bottles is 9 times the number of minutes. The number of bottles is $b$ and the number of minutes is $t$ , so one equation says this for all the rows at once: $b = 9t$ .

An equation with two variables in it is called a two-variable equation. In this lesson, you write it with the dependent variable alone on the left.

To write a two-variable equation from a table, read across each row. A row is one pair: the independent value in the first column and its dependent value in the second. Each rule in this lesson multiplies by the same number every time or adds the same number every time, so run two checks.

Subtract check: in each row, subtract the independent value from the dependent value. If the difference is the same in every row, then add that difference fits every row.
Divide check: in each row, divide the dependent value by the independent value. This needs independent values that are not 0. If the quotient is the same in every row, then multiply by that quotient fits every row.

A rule counts only if it fits every row. A table shows only some pairs, so the situation must tell you that the same rule keeps going. The next table shows what it costs to rent a kayak.

Kayak table: h is the number of hours rented, c is the total cost in dollars

Hours rented, h	Total cost in dollars, c
$2$	$14$
$3$	$21$
$4$	$28$
$5$	$35$

Worked example

Writing an equation from a table

The kayak table above shows the cost of renting a kayak. Every hour costs the same amount. Let $h$ be the number of hours you rent the kayak. Let $c$ be the total cost in dollars. Write a two-variable equation that gives $c$ from $h$ .

Say what the table counts and name the roles. The first column counts hours and the second counts dollars. Each row is one rental. You choose the hours, and the cost depends on them.
$\text{independent: } h \qquad \text{dependent: } c$
Run the subtract check across each row: dependent value minus independent value. The differences are not the same, so no rule that adds the same number fits every row.
$14 - 2 = 12, \quad 21 - 3 = 18, \quad 28 - 4 = 24, \quad 35 - 5 = 30$
Run the divide check across each row: dependent value divided by independent value. The quotient is 7 in every row, so multiplying by 7 fits every row.
$14 \div 2 = 7, \quad 21 \div 3 = 7, \quad 28 \div 4 = 7, \quad 35 \div 5 = 7$
Write the rule with variables, with the dependent variable alone on the left. The cost is 7 times the hours, and you write 7 times h as 7h. The 7 is the cost in dollars of one hour.
$c = 7h$
The divide check already showed that every row fits. See the equation at work on one row: replace h with 5, and the equation gives 35 dollars, as in the table.
$c = 7 \times 5 = 35$

Answer

The equation is $c = 7h$ . The total cost in dollars is $7$ times the number of hours.

Quiz table: q is the points from questions, s is the final score in points

Points from questions, q	Final score in points, s
$3$	$6$
$5$	$8$
$9$	$12$
$12$	$15$

Worked example

When one row fits two rules

In a quiz game, every team gets the same number of bonus points added to the points it earns from questions. The quiz table above shows four teams. Let $q$ be the points a team earns from questions. Let $s$ be the team's final score in points. Write a two-variable equation that gives $s$ from $q$ .

Say what the table counts and name the roles. The first column counts points from questions and the second counts final scores. Each row is one team. A team's question points come first, and its final score depends on them.
$\text{independent: } q \qquad \text{dependent: } s$
Look across the first row only. Two rules fit it: add 3, or multiply by 2. One row cannot tell you which rule is right, so test the other rows too.
$3 + 3 = 6, \quad 3 \times 2 = 6$
Run the divide check. The first row gives 2, but the second row gives 1.6. The quotients are not the same, so no rule that multiplies by the same number fits every row.
$6 \div 3 = 2, \quad 8 \div 5 = 1.6$
Run the subtract check across every row. The difference is 3 each time, so adding 3 fits every row. The 3 is the bonus that each team gets.
$6 - 3 = 3, \quad 8 - 5 = 3, \quad 12 - 9 = 3, \quad 15 - 12 = 3$
Write the rule with variables, with the dependent variable alone on the left. You could also write 3 + q on the right. Adding in either order gives the same sum, so both forms state the same rule.
$s = q + 3$

Answer

The equation is $s = q + 3$ . A team's final score is its points from questions plus a bonus of $3$ points.

The big idea: when one quantity depends on another, one rule links the two. The independent variable stands for the quantity whose value comes first. The dependent variable stands for the quantity whose value depends on it.

You can now show that rule in two ways. A table shows it one row at a time: say what each column and each row counts, then apply the rule to complete the table. A two-variable equation states it once: read across the rows, find the rule that fits every row, and put the dependent variable alone on the left.

Next, you will write a two-variable equation straight from a word problem, use an equation to find a value, and explain how the dependent variable changes.

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Two-Variable Equations: Write, Use, and Explain Change

16 min read

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