Introduction to Algebra

Beyond Known Numbers

In arithmetic, you work with numbers you know. For example, $3 + 5 = 8$. You start with known values and find a definite answer. Algebra takes a step into the unknown. It's a powerful tool for describing situations where a value might change or isn't known yet.

Think of it like a placeholder. If you're planning a party and don't know how many guests will come, you can't calculate the exact cost. But you can create a formula that works no matter how many people show up. That's where algebra begins, with the idea of a variable.

variable

noun

A symbol, usually a letter, that represents an unknown or changing value in a mathematical expression.

The most common letters for variables are $x$ and $y$ , but you can use any letter. Sometimes, it helps to pick a letter that reminds you of what it represents, like using $c$ for cost or $h$ for hours.

For example, if you earn 💲20 per hour, you can represent your total earnings for some number of hours as $20h$ . Here, $h$ is the variable because the number of hours you work can change.

Building Expressions

When we combine variables with numbers and arithmetic operations (+, -, ×, ÷), we create an algebraic expression. The numbers in these expressions that stand on their own are called constants, because their value is fixed.

constant

noun

A value in an expression that does not change.

Let's look at an expression: $4x - 7$.

In this expression, $x$ is the variable. The number $4$ is called the coefficient of $x$. It's the number being multiplied by the variable. The number $-7$ is the constant. An algebraic expression is a phrase, not a full sentence. It doesn't have an equals sign, so we can't "solve" it. Instead, we can evaluate it or simplify it.

Expression	Variable(s)	Coefficient(s)	Constant(s)
$5x + 2$	$x$	$5$	$2$
$y - 8$	$y$	$1$	$-8$
$2a + 3b$	$a, b$	$2, 3$	None
$10 - z$	$z$	$-1$	$10$

Simplifying Expressions

Sometimes, expressions look more complicated than they need to be. We can often simplify them by combining "like terms." Like terms are terms that have the exact same variable part.

Think of it like sorting fruit. You can combine 3 apples and 2 apples to get 5 apples. But you can't combine 3 apples and 4 bananas to get 7 "apple-bananas." In the same way, you can combine $3x$ and $2x$ to get $5x$, but you can't combine $3x$ and $4y$.

Let's simplify the expression $7x + 2 - 4x + 5$.

First, identify the like terms. The terms with $x$ are $7x$ and $-4x$. The constant terms are $2$ and $5$.

Now, combine them:

Combine the $x$ terms: $7x - 4x = 3x$
Combine the constants: $2 + 5 = 7$

The simplified expression is $3x + 7$. It's a much cleaner way to represent the same mathematical idea.

\begin{aligned} & \quad \, 7x + 2 - 4x + 5 \\ &= (7x - 4x) + (2 + 5) \\ &= 3x + 7 \end{aligned}

Here's another one: $2a + 3b - a + b$.

The like terms are $2a$ and $-a$, and the other pair is $3b$ and $b$. Remember that $b$ is the same as $1b$.

Combining them gives us: $(2a - a) + (3b + b) = a + 4b$. Since $a$ and $b$ are different variables (apples and bananas), we can't simplify this any further.

Mastering these basics—understanding variables, constants, and how to combine them—is the first crucial step. It gives you the language to start exploring the more powerful aspects of algebra.

Ready to test your understanding?

Quiz Questions 1/5

What is the primary role of a variable in algebra?

Quiz Questions 2/5

In the algebraic expression $4x - 7$ , what is the term for the number 4?

You've now learned the building blocks of algebra. These concepts are the foundation for solving problems and describing relationships in the world around us.

Algebra Essentials