Lesson 1 – Variables and Expressions


Goals

  • Learn about variables
  • Learn about algebraic expressions
  • Learn about algebraic operators
  • Learn to write mathematical expressions containing variables

What is a variable?

A variable is a value that may change. A variable can be represented by a letter such as X or Y. The letter forms a placeholder for an actual number that we don’t know yet, and X or Y can potentially be numbers like 1, 2, 3 etc.


What is an algebraic expression?

An algebraic expression is formed of one or more numbers, variables, and algebraic operators.

An example of an algebraic expression is 2*x. Another example of an algebraic expression is x+5. See how we combined numbers and variables with the algebraic operators?


What is an algebraic operator?

An algebraic operator is something that is performed on an algebraic variable, term or expression. There are six algebraic operators: Addition, Subtraction, Multiplication, Division, Exponentiation, and Roots.

Here are examples of each algebraic operator:

Addition
2 + 3 = 5
5+10 = 15

Subtraction
10 – 2 = 8
3 – 2 = 1

Multiplication
3 x 4 = 12
5 x 5 = 25

Division
8 / 2 = 4
10 / 5 = 2

Exponentiation
22 = 4
23 = 8

Roots
 9  = 3
 16  = 4


How do I write mathematical expressions containing variables?

It’s important to note that instead of using an x for multiplication, we will now be using *. Therefore, 3 x 4 = 12 is now written as 3 * 4 = 12 to reduce confusion whether x is multiplication or the variable X.

You write mathematical expressions by combining numbers, algebraic operators, and variables. This results in something that can be “evaluated”, or solved for if given a value for the variable.

Question: How would you write “Two times X plus three” ?
Answer 1: 2*x+3
Answer 2: 2x+3

When a number prefixes a variable we assume that they are multiplied together. 4x means “four times x” and .5x means “.5 times x” which equals one half of x.

Exponentiation is also known as powers, because 23 is read as “two to the third power”. Two would be the base, and three would be the exponent. This is like multiplying 2 * 2 * 2 (note there are three twos!). If it was 24 it would be 2 * 2 * 2 * 2. If it were 32 it would be 3 * 3. If it was 33 it would be 3 * 3 *3, and finally if it was 34 it would be 3 * 3 * 3 * 3.

For our purposes now we will only focus on taking square roots, but keep in mind that you can take 3rd roots and more. Taking a square root is the same as asking “Which number can I multiply by itself to give us the current number?”. So for example, the square root of 9 is 3 because 3 times itself is 9.

Question: What do we ask ourselves when taking the square root of 16?
Answer: “What number times itself equals 16?”

4 times itself is 16; 4 * 4 = 16. This means that 4 is the square root of 16.

Reviewing Variables and Expressions

At this point, you should know what a variable is, what an algebraic expression is and what they represent, the six types of algebraic operators, and how to write mathematical expressions containing variables.

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    1. Lesson 2 – Order of Operations

      [...] Order of operations is the sequence in which you evaluate the math for algebraic operators. Don’t forget to review the 6 algebraic operators. [...]

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