Variables in Algebra

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In this lesson, you will learn

  • What a variable is and why we use letters for unknown numbers
  • How to find the value of a variable and replace the variable with it
  • The difference between a variable with one value and variables that can change
  • How to graph a variable on a number line

What Is a Variable in Algebra?

Definition

Variables in algebra are letters, such as x, y, v or g (or any letter you want), that can be used to represent unknown numbers.

In C = 2Ο€r, the letters C and r are labeled variables. In 3x βˆ’ 4 = 11, the letter x is labeled variable.

Basically, whenever we do not know the value of a number, we use a variable until we figure out what the number is. The variables that we often use are x and y, with x perhaps being the most popular.

Once the value of the variable is known, you can replace the variable or the letter with the known value.

Common mistake

Reading 3x as "3 plus x". A number written right next to a variable means multiplication, so 3x means 3 times x. If x = 5, then 3x = 3 Γ— 5 = 15, not 35 or 8.

In the equation 7y + 2 = 30, which letter or number is the variable?

Show answer

y is the variable. It is the letter that stands for the unknown number. The numbers 7, 2 and 30 are known.

A Variable With One Value: 3x βˆ’ 4 = 11

In the figure above, x is a variable in the equation 3x βˆ’ 4 = 11 because we have no idea what the value of x is at the moment.

Let us figure it out.

3x βˆ’ 4 = 11

3x βˆ’ 4 + 4 = 11 + 4

3x = 15

x = 153

x = 5

Now that we know the value of x, we can replace it in the equation with 5.

Worked example: Replace x with 5 in 3x βˆ’ 4 = 11. Is the equation true?
  1. Write 5 in place of x: 3 Γ— 5 βˆ’ 4 = 11.
  2. Multiply first: 3 Γ— 5 = 15, so the left side is 15 βˆ’ 4.
  3. Subtract: 15 βˆ’ 4 = 11.
  4. Both sides are 11, so the equation is true.

Answer: yes, x = 5 makes 3x βˆ’ 4 = 11 true.

Notice that x represents a variable that has a value that cannot change.

x will always equal 5 for the equation 3x βˆ’ 4 = 11.

What is the value of x in x βˆ’ 6 = 9?

Show answer

x = 15. Add 6 to both sides: x βˆ’ 6 + 6 = 9 + 6, so x = 15. Check: 15 βˆ’ 6 = 9.

Variables That Can Change: C = 2Ο€r

You may have recognized the equation C = 2Ο€r. It is the equation that we use to find the circumference of a circle.

This equation has two variables, and these are r and C.

Unlike x in the equation 3x βˆ’ 4 = 11 that has a value that cannot change, r and C can take on values that can change.

This makes sense since the radius of the circle (r) can be any length.

The value of the circumference (C) will change based on the value of r.

Remember

Ο€ is a letter, but it is not a variable. It stands for one number (about 3.14) that never changes.

In C = 2Ο€r, what happens to C when the radius r gets bigger?

Show answer

C gets bigger. C is 2Ο€ times r, so a bigger radius gives a bigger circumference.

How to Graph Variables in Algebra

Once you know the value of a variable, you can graph it, since all numbers can be graphed on a number line. Just locate the value on the number line and put a dot.

For example, graphing x = 4 gives the following graph:

Number line from βˆ’9 to 9 with a dot at 4

Other examples

Graph x = βˆ’6 on the number line.

Just locate βˆ’6 on the number line and put a dot at βˆ’6.

Number line from βˆ’9 to 9 with a dot at βˆ’6

Graph x = 0.

Just locate 0 on the number line and put a dot.

Number line from βˆ’9 to 9 with a dot at 0
Common mistake

Drawing an open circle. To graph x = 4, fill in the dot, because x is exactly 4. Open circles are used for inequalities such as x > 4, where the number itself is not included.

Your turn

Now, write a statement of equality for each of the following graphs. Use x as the variable.

Graph #1

Number line from βˆ’9 to 9 with a dot at one of the numbers
Show answer

Answer: x = βˆ’2

Graph #2

Number line from βˆ’9 to 9 with a dot at one of the numbers
Show answer

Answer: x = 8

Summary

  • A variable is a letter, such as x or y, that stands for an unknown number.
  • Once you find the value of a variable, you can replace the letter with that value.
  • In an equation like 3x βˆ’ 4 = 11, x has one value that cannot change: x = 5.
  • In a formula like C = 2Ο€r, the variables r and C can take on many values.
  • To graph a variable, locate its value on a number line and put a dot there.

Test Yourself: 10-Question Quiz

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