Introduction to Algebra

The Mystery of the Letter

Arithmetic is all about numbers you know, like $2 + 2 = 4$. Algebra starts when we introduce numbers we don't know. To do this, we use letters to stand in for these mystery values.

Variable

noun

A symbol, usually a letter, that represents a number or value that can change or is unknown.

Think of a variable as a labeled box. We might not know what's inside the box labeled 'x', but we know it holds a number. We can still do math with it. For example, if you have a box 'x' and you add 5 to it, you have $x + 5$ . The total depends on what number is eventually put into the box.

Building with Expressions

When we combine variables, numbers, and operations like +, –, ×, and ÷, we create an algebraic expression. It's like a mathematical phrase. Here are a few examples:

$a + 10$
$4y - 3$
$2x + z$

These expressions have a few key parts: variables (the letters), constants (the numbers that stand alone), and operators (the math symbols). In $4y - 3$ , the variable is $y$ , the constant is $-3$ , and the operator is –.

Sometimes, we can tidy up expressions by combining similar things. This is called simplifying. We can only combine "like terms"—terms that have the exact same variable parts.

Imagine you have a bag with 2 apples, 3 oranges, and then someone adds another apple. You now have 3 apples and 3 oranges. You can't combine apples and oranges. In algebra, it's the same. If we have the expression $2x + 3y + x$ , we can combine the $x$ terms.

$2x + x$ gives us $3x$ . The $3y$ is different, so we leave it alone.

The simplified expression is $3x + 3y$ .

Equations and Balance

An equation is a statement that two expressions are equal. You can always spot an equation by its equals sign (=). Think of it as a perfectly balanced scale. What's on the left side has the exact same value as what's on the right.

The equation $x + 3 = 7$ states that the expression on the left ( $x+3$ ) is equal to the expression on the right (7). The goal in algebra is often to figure out the value of the variable that makes this statement true. In this case, $x$ must be 4 to keep the scale balanced.

When Things Aren't Equal

Sometimes we don't know that two things are exactly equal. We might only know that one is bigger or smaller than the other. This is where inequalities come in. Instead of an equals sign, they use different symbols to compare values.

Symbol	Meaning	Example	What it Says
$>$	Greater than	$x > 5$	$x$ is any number bigger than 5.
$<$	Less than	$y < 12$	$y$ is any number smaller than 12.
$\ge$	Greater than or equal to	$z \ge 20$	$z$ can be 20, or any number bigger.
$\le$	Less than or equal to	$a \le 100$	$a$ can be 100, or any number smaller.

Unlike an equation which might have one specific solution, an inequality describes a whole range of possible values for the variable. For $x > 5$ , the variable could be 6, 5.1, or 9000. There are infinite possibilities!

Now, let's test your understanding of these building blocks.

Quiz Questions 1/5

In the algebraic expression $6y - 11$ , what is the variable?

Quiz Questions 2/5

Which of the following is an equation?

These concepts of variables, expressions, equations, and inequalities are the foundation of all algebra. Mastering them opens the door to solving a huge range of interesting problems.