Two-Variable Equations: Write, Use, and Explain Change | Grade 6 Math | LevelKite

Some word problems describe two quantities that change together. Think of a printer that prints 12 pages each minute. The number of minutes can change, and the number of pages changes with it.

You already know the words for these two roles. A variable is a letter that stands for the value of a quantity that can change. The independent variable stands for the quantity you choose or that changes on its own. The dependent variable stands for the quantity whose value depends on it. You have also written an equation from a table of values.

Now the same kind of relationship comes to you in sentences. In this lesson you will do three things:

write an equation straight from the words of a problem,
use that equation to find a value you do not know,
explain how the dependent variable changes when the independent variable changes.

An equation says that two amounts are equal. A two-variable equation is an equation with two variables. It shows how the value of one quantity is tied to the value of another.

Each word problem in this lesson gives you two quantities that change and one number that stays fixed. To turn the words into an equation, make four moves:

Name the two quantities and their units, such as minutes or pounds. Choose a letter for each one. Each letter stands for a number.
Decide which quantity depends on the other. That one is the dependent variable. Many problems tell you by asking for an equation that gives one quantity from the other. The quantity the equation gives is the dependent one.
Find the rule. The rule is what you do to the independent variable to get the dependent variable. Try small numbers first, as if you were filling in rows of a table.
Write the equation with the dependent variable alone on one side. Then test it with a pair of numbers from the story.

In this lesson every rule has one step. You either multiply the independent variable by a fixed positive number, or you add a fixed positive number to it.

Worked example

Write an equation with a multiplication rule

A printer prints 12 pages each minute. Write an equation that gives the number of pages printed from the number of minutes the printer runs.

Name the two quantities that change and choose a letter for each. Let m be the number of minutes the printer runs. Let p be the number of pages it prints. The 12 pages each minute is the number that stays fixed.
$m = \text{number of minutes}, \quad p = \text{number of pages}$
Decide which quantity depends on the other. The printer prints more pages when it runs longer, so the number of pages depends on the number of minutes. That makes m the independent variable and p the dependent variable.
Find the rule with small numbers. In 1 minute the printer prints 12 pages. In 2 minutes it prints 2 groups of 12 pages. In 3 minutes it prints 3 groups of 12 pages. Each time, the number of pages is 12 times the number of minutes.
$12 \times 1 = 12, \quad 12 \times 2 = 24, \quad 12 \times 3 = 36$
Write the same rule with the variables. Put the dependent variable p alone on one side, because p is the result of the rule. Writing 12 next to m means 12 times m.
$p = 12m$
Check the equation with a pair of values you already worked out. In 3 minutes the printer prints 36 pages. Put 3 in place of m and see that the equation gives 36.
$p = 12 \times 3 = 36$

Answer

The equation is $p = 12m$ , where $m$ is the number of minutes the printer runs and $p$ is the number of pages it prints.

Worked example

Write an equation with an addition rule

An empty backpack weighs 2 pounds. Write an equation that gives the total weight of the packed backpack from the weight of the books inside it.

Name the two quantities that change, with their units. Let b be the weight of the books in pounds. Let t be the total weight of the packed backpack in pounds. The 2 pounds for the empty backpack stays fixed.
$b = \text{pounds of books}, \quad t = \text{total pounds}$
Decide which quantity depends on the other. The total weight depends on how many pounds of books go in. So b is the independent variable and t is the dependent variable.
Find the rule with small numbers. With 3 pounds of books, the total is 3 plus 2. With 6 pounds of books, the total is 6 plus 2. Each time, you add 2 to the weight of the books.
$3 + 2 = 5, \quad 6 + 2 = 8$
Write the rule with the variables, keeping the dependent variable t alone on one side. Then check it with a pair from the last step. Books that weigh 6 pounds should give a total of 8 pounds.
$t = b + 2, \qquad 6 + 2 = 8$

Answer

The equation is $t = b + 2$ , where $b$ is the weight of the books in pounds and $t$ is the total weight of the packed backpack in pounds.

A correct equation is true for every pair of values that fits the story. With the one-step rules in this lesson, knowing the value of one variable lets you find the value of the other.

To substitute means to put a number in place of a variable. Substitute the value you know. Then work out the other value.

If you know the independent variable, you only need to calculate. In these equations the dependent variable is already alone on one side.
If you know the dependent variable, you work backward with an inverse operation. An inverse operation undoes another operation. Dividing by a number that is not zero undoes multiplying by that number. Subtracting a number undoes adding that number.

When you work backward, do the same thing to both sides of the equation. The two sides are equal amounts. If you divide both by the same number that is not zero, or subtract the same number from both, they stay equal. So the equation still describes the same relationship.

Worked example

Find a value in each direction

Use the printer equation $p = 12m$ , where $m$ is the number of minutes the printer runs and $p$ is the number of pages it prints. (a) How many pages does it print in 15 minutes? (b) How many minutes does it take to print 84 pages?

Part (a): you know the independent variable. The printer runs for 15 minutes, so substitute 15 for m.
$p = 12 \times 15$
Multiply. Split the 15 minutes into 10 minutes and 5 minutes, because 15 groups of 12 are 10 groups of 12 and 5 more groups of 12. Ten groups of 12 are 120. Five groups of 12 are 60. Add the two parts.
$12 \times 15 = 12 \times 10 + 12 \times 5 = 120 + 60 = 180$
Part (b): you know the dependent variable. The printer prints 84 pages, so substitute 84 for p.
$84 = 12m$
In this equation m is multiplied by 12. Dividing by 12 undoes multiplying by 12, so divide both sides by 12. The sides are equal amounts, and you do the same thing to each, so they stay equal.
$84 \div 12 = 12m \div 12$
Work out each side. On the right, multiplying m by 12 and then dividing by 12 leaves just m. On the left, 84 divided by 12 is 7. So m is 7.
$7 = m$
Check by putting 7 in place of m in the original equation. Seven groups of 12 are 7 tens and 7 twos, which is 70 plus 14. You get 84 pages, so the answer fits.
$12 \times 7 = 70 + 14 = 84$

Answer

The printer prints $180$ pages in $15$ minutes, and it takes $7$ minutes to print $84$ pages.

Now look at how the two quantities move together. To explain a change, compare two values of the independent variable. Find the dependent variable for each one. Then see how far apart the two results are.

A table lets you compare several values at once. In the table below, each row is one running time for the printer. The first column counts the minutes the printer runs. The second column shows the multiplication from the equation $p = 12m$ . The third column counts the pages printed in that time.

Each row is one running time for the printer.

Minutes the printer runs (m)	Work: 12 times m	Pages printed (p)
$1$	$12 \times 1$	$12$
$2$	$12 \times 2$	$24$
$3$	$12 \times 3$	$36$
$4$	$12 \times 4$	$48$
$5$	$12 \times 5$	$60$

Read down the first column. The number of minutes, the independent variable, goes up by 1 from each row to the next. Now read down the last column: $12$ , $24$ , $36$ , $48$ , $60$ . The number of pages, the dependent variable, goes up by 12 each time. Another word for goes up is increases. You can check with subtraction: $24 - 12 = 12$ and $36 - 24 = 12$ .

This is not luck. One more minute means one more group of 12 pages. So the fixed number 12 in $p = 12m$ is the amount that $p$ gains for each extra minute. In the same way, one minute less means 12 pages fewer.

A bigger jump works the same way. A jump of 3 minutes adds 3 groups of 12. From the row for 2 minutes to the row for 5 minutes, the pages go from $24$ to $60$ , and $60 - 24 = 36$ , which is $12 \times 3$ .

Notice the wording. From one row to the next, $p$ goes up by 12. That is adding 12, not multiplying by 12.

You can show the same pattern without a table. Suppose the minutes go from $m$ to $m + 1$ . Then the pages go from $12m$ to $12(m + 1)$ . That is $m + 1$ groups of 12 pages, which is the same amount as $m$ groups of 12 plus one more group of 12. Both ways of writing it name the same number of pages for every value of $m$ . So the new number of pages is always the old number plus 12:

12(m + 1) = 12m + 12

Worked example

Explain a change with an addition rule

Use the backpack equation $t = b + 2$ , where $b$ is the weight of the books in pounds and $t$ is the total weight of the packed backpack in pounds. The weight of the books goes up from 4 pounds to 7 pounds. Explain how the total weight changes.

Find the total weight at the start. There are 4 pounds of books, so substitute 4 for b.
$t = 4 + 2 = 6$
Find the total weight at the end. There are now 7 pounds of books, so substitute 7 for b.
$t = 7 + 2 = 9$
Compare the two values of each variable. Subtract to find how much each one went up. The books went up by 3 pounds, and the total went up by 3 pounds.
$7 - 4 = 3, \qquad 9 - 6 = 3$
Explain why the two changes match. The new total is 4 plus 3 plus 2. You can add in any order, so that is the old total, 4 plus 2, with 3 more. The 2 pounds for the empty backpack is added both times, so only the extra books change the total.
$(4 + 3) + 2 = (4 + 2) + 3 = 6 + 3 = 9$

Answer

When the weight of the books goes up by $3$ pounds, the total weight also goes up by $3$ pounds, from $6$ pounds to $9$ pounds.

Key concept

An equation is the rule that links two quantities

A two-variable equation tells you how to get the dependent variable from the independent variable. Write it from the words of the problem. Then use it to find a value and to explain a change.

The rule also tells you how the dependent variable changes. Here $x$ is the independent variable and $y$ is the dependent variable:

Multiplication rule, $y = kx$ , where $k$ is a fixed positive number: each time $x$ goes up by 1, $y$ goes up by $k$ .
Addition rule, $y = x + c$ , where $c$ is a fixed positive number: when $x$ goes up by some amount, $y$ goes up by the same amount.

These patterns belong to these two kinds of rule. For any other rule, compare two values to find the change.

Takeaway. A word problem about two changing quantities has a rule inside it. Name each quantity with its unit, decide which one depends on the other, and write the rule as an equation with the dependent variable alone on one side. Test the equation with numbers you can work out from the story.

After that, the equation does the work. Substitute a value you know to find the one you do not know. Use an inverse operation on both sides when you have to work backward. To explain a change, compare two values of the independent variable and see what happens to the dependent variable.