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3.1: One-Step Equations

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Learning Objectives

  • Solve an equation using addition.
  • Solve an equation using subtraction.
  • Solve an equation using multiplication.
  • Solve an equation using division.


Nadia is buying a new mp3 player. Peter watches her pay for the player with a $100 bill. She receives $22.00 in change, and from only this information, Peter works out how much the player cost. How much was the player?

In algebra, we can solve problems like this using an equation. An equation is an algebraic expression that involves an equals sign. If we use the letter x to represent the cost of the mp3 player, we can write the equation x + 22 = 100. This tells us that the value of the player plus the value of the change received is equal to the $100 that Nadia paid.

Another way we could write the equation would be x = 100 - 22. This tells us that the value of the player is equal to the total amount of money Nadia paid (100 - 22). This equation is mathematically equivalent to the first one, but it is easier to solve.

In this chapter, we will learn how to solve for the variable in a one-variable linear equation. Linear equations are equations in which each term is either a constant, or a constant times a single variable (raised to the first power). The term linear comes from the word line, because the graph of a linear equation is always a line.

We’ll start with simple problems like the one in the last example.

Solving Equations Using Addition and Subtraction

When we work with an algebraic equation, it’s important to remember that the two sides have to stay equal for the equation to stay true. We can change the equation around however we want, but whatever we do to one side of the equation, we have to do to the other side. In the introduction above, for example, we could get from the first equation to the second equation by subtracting 22 from both sides:

 x + 22 &= 100\\x + 22 - 22 &= 100 - 22\\x &= 100 - 22

Similarly, we can add numbers to each side of an equation to help solve for our unknown.

Example 1

Solve x - 3 = 9.


To solve an equation for x, we need to isolate x-that is, we need to get it by itself on one side of the equals sign. Right now our x has a 3 subtracted from it. To reverse this, we’ll add 3—but we must add 3 to both sides.

 x - 3 &= 9\\x - 3 + 3 &= 9 + 3\\x + 0 &= 9 + 3\\x &= 12

Example 2

Solve z - 9.7 = -1.026


It doesn’t matter what the variable is—the solving process is the same.

 z - 9.7 &= -1.026\\ z - 9.7 + 9.7 &= -1.026 + 9.7\\z &= 8.674

Make sure you understand the addition of decimals in this example!

Example 3

Solve x + \frac{4}{7} = \frac{9}{5}.


To isolate x, we need to subtract \frac{4}{7} from both sides.

 x + \frac{4}{7} &= \frac{9}{5}\\x + \frac{4}{7} - \frac{4}{7} &= \frac{9}{5} - \frac{4}{7}\\x &= \frac{9}{5} - \frac{4}{7}

Now we have to subtract fractions, which means we need to find the LCD. Since 5 and 7 are both prime, their lowest common multiple is just their product, 35.

x &= \frac{9}{5} - \frac{4}{7}\\x &= \frac{7 \cdot 9}{7 \cdot 5} - \frac{4 \cdot 5}{7 \cdot 5}\\x &= \frac{63}{35} - \frac{20}{35}\\x &= \frac{63 - 20}{35}\\x &= \frac{43}{35}

Make sure you’re comfortable with decimals and fractions! To master algebra, you’ll need to work with them frequently.

Solving Equations Using Multiplication and Division


Example 4




Example 5

Solve  \frac{9x}{5} = 5.



Example 6

Solve 0.25x = 5.25.




Example 7

Solve 5x = 3.25.

To cancel the 5, we divide both sides by 5.


Example 8

Solve 7x = \frac{5}{11}.

Divide both sides by 7.


Example 9

Solve 1.375x = 1.2.

Divide by 1.375




Solve Real-World Problems Using Equations

Example 10

In the year 2017, Anne will be 45years old. In what year was Anne born?



Anne was born in 1972.

Example 11


How much does each DVD player weigh?



Each DVD player weighs 2.5 pounds.

Example 12


a) How many minutes it took him to eat one hot dog.

b) How many hot dogs he ate per minute.

c) What his old record was.




b) Now we’re looking for hot dogs per minute instead of minutes per hot dog. We’ll use the variable y instead of x this time so we don’t get the two confused. 12 minutes, times the number of hot dogs per minute, equals the total number of hot dogs, so 12y = 53.5. Dividing both sides by 12 gives us y = \frac{53.5}{12}, or y = 4.458 hot dogs per minute.


Lesson Summary


Review Questions

  1. Solve the following equations for x.
    1. x = 11 = 7
    2. x - 1.1 = 3.2
    3. 7x = 21
    4. 4x = 1
    6. x + \frac{5}{2} = \frac{2}{3}
    7. x - \frac{5}{6} = \frac{3}{8}
    8. 0.01x = 11
  2. Solve the following equations for the unknown variable.
    1. q - 13 = -13
    2. z + 1.1 = 3.0001
    3. 21s = 3
    4. t + \frac{1}{2} = \frac{1}{3}
    6. \frac{3}{4} = -\frac{1}{2} - y
    7. 6r = \frac{3}{8}
    2. How many tokens he collects per week, w.
    1. How many cookies will she make in all?
    2. How many cups of flour go into one batch?
    3. How many eggs go into one batch?

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