Introduction to Algebra

Algebra is all about solving problems. It’s a branch of mathematics where we commonly use letters (and sometimes symbols) to represent numbers in equations or expressions. These letters or symbols are called variables and they stand-in for numbers that we haven't solved yet. Don’t worry — it’s not as tricky as it sounds. By the end of this lesson, you’ll understand the basics and learn to solve some algebraic expressions yourself.

What is a Variable?

In algebra, we often use letters like x, y, or z to stand for numbers. A variable is like a box with a mystery number inside. For example:

  • If we say x = 5, the variable x is holding the value 5.
  • If we don’t know the value of x, we can still write it in equations and work to figure it out, or we may use it to understand relationships between larger expressions.

Example:

Let’s say we have this equation:

$$ x + 1 = 4 $$

This means "what number added to 1 gives us 4?" If you think about it for a moment, you'll recall that 3 is what we add to 1 to get a total of 4. In this case, x=3.

Expressions vs. Equations

Expressions

Let's discuss two algebra terms quickly - expression and equation. An expression is a collection of numbers, variables, and operations (like +, −, ×, ÷) written in a mathematical way. 4+3 is an expression. x+1 is an expression. They have meaning, but don't convey a full problem to be solved yet. Expressions have to be combined with other expressions and an equals sign (=) to form equations.

Examples of expressions:

  • 1 + 1
  • 3x + 2
  • 5y - 7
  • 2a + b

Equations

An equation has an equals sign (=) and shows a relationship between two sides, each of which is one or more expressions. A fundamental concept of algebra is that both sides of an equals sign are - wait for it - the same! The complicated math on each side must balance, like two sides of a scale.

Examples of equations:

  • 3x + 2 = 11
  • 5x - 7 = 8
  • 2a + b = 10

When solving or simplifying equations, we may be directed to find the value of the variable that makes the equation true. In order to do that, we can make changes to each side as long as we do the same to each side, like adding or removing the same weight from both sides of a balanced scale.

Writing Algebraic Expressions

Let’s turn real-world problems into algebraic expressions!

Example:

Problem: A farmer raises chickens. Each chicken lays 2 eggs per day. Write an expression for the total number of eggs laid by the chickens in a day.

Solution: Let c be the number of chickens. Each chicken lays 2 eggs per day, so the total number of eggs each day is: 2c

Example:

Problem: Sam has $10, and he spends x dollars on candy at Halloween. Write an expression for the money Sam has left.

Solution: Sam starts with $10 and spends $x, so the expression for the leftover money is: 10 - x

Solving Simple Equations

When we solve an equation, we figure out the value (or sometimes values!) of the variable that makes it true (makes the equation balance).

Example:

Problem: Solve for x: x + 5 = 12

Solution: To find x, subtract 5 from both sides of the equation. Remember, as long as we do the same thing to each side, adding or subtracting the same amount, for example, we will still have an balance equation. By subtracting 5 from each side we can get the x by itself and math will show us what x equals:

  • x + 5 = 12
  • x + 5 (- 5) = 12 (- 5)
  • x + 0 = 7
  • x = 7

Collecting Like Terms

Sometimes we have more than one term in an equation that contains the variable we wish to solve for. We can collect like terms to combine all of the terms with the same variable.

We'll learn a little later that collecting like terms means combining terms that have the same variable raised to the same power. Like terms are terms that share the same variable and exponent, although their coefficients may be different. For now, we'll leave the exponent alone and stay simple. Let me show you what I mean:

In the expression 3x + 5x, both terms are "like terms" because they have the same variable x. To combine them, simply add or subtract their coefficients, the numbers in front: 3x + 5x = 8x. However, terms with different variables, such as 2x and 3y, cannot be combined because they are not like terms. Collecting like terms simplifies expressions and makes it easier to work with them.

Practice Problems

Now it’s your turn! Try these practice problems.

Simplify Each Expression:

  1. 3x + 4x
  2. 2a + 5 - a

Solve Each Equation:

  1. x - 3 = 10
  2. 2y + 5 = 13

Real-World Problem:

A school cafeteria serves x students every day. Each student gets 3 slices of pizza. Write an expression for the total slices of pizza needed. If x = 30, how many slices are necessary?

Now you've got a basic understanding of algebra, and can solve equations like \(7+x=14\). If not, you might want to re-read the lesson, browse some other algebra lessons, or ask for help.