Chapter 6 · Equations and inequalities

Writing and Solving Two-Step Equations

Turn a word problem into a two-step equation, then solve it by undoing the operations in reverse order and checking the answer.

18 min read

A climbing gym charges 9 dollars for each visit, plus a one-time fee of 12 dollars for a membership card. Maya has paid 57 dollars in all. How many visits has she made?

This story has two parts: an amount that repeats (9 dollars for every visit) and an amount that happens once (the 12-dollar card). In this lesson you'll turn stories like this into equations and then solve them. You already know how to solve one-step equations. Here you'll use two of those steps, one after the other.

Before using a letter, try a few numbers of visits and watch what stays the same.

Total cost for different numbers of visits

Number of visits	Calculation (dollars)	Total cost (dollars)
$1$	$9(1) + 12$	$21$
$2$	$9(2) + 12$	$30$
$3$	$9(3) + 12$	$39$
? (not known yet)	$9(?) + 12$	$57$

Read down the middle column. Every calculation has the same shape: 9 times the number of visits, then add 12. Only the number of visits changes, so give it a letter. A variable is a letter that stands for a number. Let $v$ be the number of visits Maya has made.

With $v$ visits, the cost in dollars is $9v + 12$ , where $9v$ means $9 \times v$ . That is an expression: a mathematical phrase with no equals sign. Its $9v$ part grows with every visit, and its 12 stays the same.

Maya paid 57 dollars, so the expression must equal 57. An equation is a statement that two expressions are equal. Here is Maya's equation, with each part labeled:

\underbrace{9v}_{\text{cost of } v \text{ visits}} + \underbrace{12}_{\text{card fee}} = \underbrace{57}_{\text{total paid}}

Maya's equation is a two-step equation: an equation in which two operations, such as multiplying and adding, are done to the variable. In this lesson, each one has the form $px + q = r$ . You multiply the variable $x$ by a known number $p$ , then add a known number $q$ , and the result is the known number $r$ . Subtracting fits this form too. Subtracting 8, for example, is the same as adding $-8$ , so then $q = -8$ .

The number that multiplies the variable, $p$ , is called the coefficient. It cannot be zero, or the variable would disappear. In Maya's equation the variable is $v$ , and $p = 9$ , $q = 12$ and $r = 57$ .

To write a two-step equation from a word problem, find four things:

The unknown. Pick a letter and say exactly what it counts.
The repeating amount. It is the amount for each one of the unknown, so it multiplies the variable.
The one-time amount. It is counted once, however large the unknown is.
The final amount. The two parts together must equal this known number.

Add a part that raises the final amount, and subtract a part that lowers it. Every part you add, subtract or set equal must be in the same unit. In Maya's equation, $9v$ , $12$ and $57$ are all amounts in dollars.

Worked example

Write an equation from a story

In a quiz game, each correct answer earns 6 points. Nia also got a one-time penalty of 8 points for starting late. Her final score was 40 points. Write an equation you could solve to find how many questions Nia answered correctly.

Name the unknown. The story asks for a number of correct answers, so let c be the number of questions Nia answered correctly.
Find the repeating amount. Each correct answer earns 6 points, so c correct answers earn 6 times c points. The 6 multiplies the variable because it is counted once for every correct answer.
$6c$
Find the one-time amount. The penalty happens once and lowers the score by 8 points, so subtract 8 from the points earned. Subtracting 8 is the same as adding negative 8, so this fits the form px + q with q equal to negative 8.
$6c - 8$
Set the expression equal to the final amount. Nia's final score was 40 points, so the points earned minus the penalty must equal 40. Every part of the equation is in points.
$6c - 8 = 40$

Answer

The equation $6c - 8 = 40$ models the story, where $c$ is the number of questions Nia answered correctly.

Now for the second skill. To solve an equation means to find each value of the variable that makes the equation true. Such a value is called a solution.

You could find Maya's visits without an equation. Work backward from 57 dollars. First take away the 12-dollar card fee, because only the rest was spent on visits. Then split what is left into 9-dollar visits.

Solving $9v + 12 = 57$ uses the same two operations in the same order. The one difference is that you do each operation to both sides. The equation says that $9v + 12$ and $57$ are two names for the same number. Do the same thing to both names and they still name one number, so the two sides stay equal. This works when you add the same number to both sides, subtract it from both sides, or multiply or divide both sides by it, as long as you never divide by zero. The table shows both ways side by side.

Two ways to find Maya's number of visits

Step	Working backward with arithmetic	Solving the equation
Start	Maya paid 57 dollars in all.	$9v + 12 = 57$
1	Take away the 12-dollar card fee. $57 - 12 = 45$ , so 45 dollars went to visits.	Subtract 12 from both sides. $9v + 12 - 12 = 57 - 12$ , so $9v = 45$ .
2	Split 45 dollars into 9-dollar visits. $45 \div 9 = 5$ , so Maya made 5 visits.	Divide both sides by 9, which is not zero. $9v \div 9 = 45 \div 9$ , so $v = 5$ .
Check	5 visits cost $9 \times 5 = 45$ dollars. With the card, $45 + 12 = 57$ dollars.	$9(5) + 12 = 45 + 12 = 57$ , so $v = 5$ makes the original equation true.

Key concept

Undo the operations in reverse order

The equation $px + q = r$ , with $p$ not zero, says: start with $x$ , multiply by $p$ , then add $q$ , and the result is $r$ . A reliable way to find $x$ is to undo those operations in reverse order.

First undo the adding or subtracting, the operation that was done last. Now $px$ is alone on its side.
Then undo the multiplying by dividing by $p$ . This is allowed because $p$ is not zero. Dividing by zero is never allowed.

In each step, do the same operation to both whole sides so that they stay equal. The value you find is a solution only if it makes the original equation true, so put it back into the original equation to check.

px + q = r \;\longrightarrow\; px = r - q \;\longrightarrow\; x = \frac{r - q}{p} \quad (p \neq 0)

Worked example

Solve a two-step equation

Solve the equation from Nia's quiz game, $6c - 8 = 40$ , where $c$ is the number of questions she answered correctly.

Read what happens to c: it is multiplied by 6, and then 8 is subtracted. Undo the last operation first. Adding 8 undoes subtracting 8, and you add it to both sides so the two sides stay equal.
$6c - 8 + 8 = 40 + 8$
Simplify each side. On the left, subtracting 8 and then adding 8 changes nothing, so the left side is equivalent to 6c. Equivalent means the two expressions are equal for every value of c. On the right, 40 plus 8 is 48.
$6c = 48$
Now c is only multiplied by 6. Dividing by 6 undoes multiplying by 6, so divide both sides by 6. This is allowed because 6 is not zero.
$\frac{6c}{6} = \frac{48}{6}$
Simplify each side. On the left, 6 times c divided by 6 is c. On the right, 48 divided by 6 is 8.
$c = 8$
Check in the original equation by putting 8 in place of c. Six times 8 is 48, and 48 minus 8 is 40, which matches the right side. So 8 makes the original equation true. It also fits the story, because 8 is a whole number of questions.
$6(8) - 8 = 48 - 8 = 40$

Answer

The solution is $c = 8$ , so Nia answered 8 questions correctly.

Watch out for dividing only part of a side. To solve $3x + 6 = 21$ for the number $x$ , you might divide by 3 right away and write $x + 6 = 7$ , which gives $x = 1$ . Check it: $3(1) + 6 = 9$ , not $21$ . So $x = 1$ does not make the original equation true.

Why does this fail? Dividing by 3 is allowed, since 3 is not zero. But the step divided only the $3x$ and left the $6$ alone, so the two sides did not get the same operation. Dividing the whole left side gives $x + 2$ , because $(3x + 6) \div 3 = x + 2$ for every value of $x$ . That leads to $x + 2 = 7$ , so $x = 5$ .

Subtracting 6 from both sides first is simpler: $3x = 15$ , then $x = 5$ . Check: $3(5) + 6 = 15 + 6 = 21$ , which is true.

Worked example

Write and solve when an amount goes down

A tank holds 50 liters of water. A valve opens, and the tank drains 4 liters each minute. After how many minutes will exactly 18 liters be left in the tank?

Let m be the number of minutes the tank has been draining. The tank loses 4 liters each minute, so after m minutes it has lost 4 times m liters. Start with 50 liters, subtract the water lost, and set the result equal to the 18 liters left. Every part of the equation is in liters.
$50 - 4m = 18$
Match the form px + q = r. Subtracting 4m is the same as adding negative 4m, and a sum can be written in either order, so the variable part can go first. The old and new left sides are equivalent: they are equal for every value of m. The coefficient is negative 4 because the amount of water goes down.
$-4m + 50 = 18$
Undo the adding of 50 by subtracting 50 from both sides, which keeps the two sides equal. The left side becomes negative 4m. On the right, 18 minus 50 is negative 32, because 50 is 32 more than 18.
$-4m + 50 - 50 = 18 - 50 \quad\text{so}\quad -4m = -32$
Undo the multiplying by dividing both sides by negative 4. This is allowed because negative 4 is not zero. On the left, negative 4 times m divided by negative 4 is m. On the right, negative 32 divided by negative 4 is 8, because negative 4 times 8 is negative 32.
$\frac{-4m}{-4} = \frac{-32}{-4} \quad\text{so}\quad m = 8$
Check in the original equation by putting 8 in place of m. Four times 8 is 32, and 50 minus 32 is 18, so 8 makes the original equation true. It also fits the story: 8 is a positive number of minutes, and water is still left in the tank.
$50 - 4(8) = 50 - 32 = 18$

Answer

The solution is $m = 8$ : exactly 18 liters are left after the tank has drained for 8 minutes.

The big idea: a two-step equation of the form $px + q = r$ tells a short story. You multiply an unknown number by one number, add or subtract a second number, and get a known result.

To write one, name the unknown, multiply it by the repeating amount, combine that with the one-time amount by adding or subtracting, and set the result equal to the final amount.
To solve one, undo the operations in reverse order. Do the same operation to both sides each time, and never divide by zero. Then check that your value makes the original equation true.

Next, you'll solve two-step equations where the variable is inside parentheses, such as $p(x + q) = r$ , and equations with fraction and decimal coefficients.

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Two-Step Equations: Parentheses, Fractions, Decimals

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