Introduction to Algebra

Algebra is the branch of math where letters and symbols stand in for numbers — usually a number you don't know yet, or one that can change depending on the situation. If you've worked with arithmetic, algebra is the next step. Instead of always using specific numbers, you'll use variables to represent unknown values.

That makes algebra useful for solving real problems: figuring out how much money you'll have after saving for a few weeks, calculating distances for a road trip, scaling a recipe up or down. Anywhere there's an unknown, algebra gives you a way to work with it.

What is a Variable?

A variable is a letter or symbol that stands in for a number you don't know yet, or one that can change. The most common variable letters are $x$, $y$, and $z$, but any letter works as long as you're clear about what it represents.

Suppose you don't know how many apples are in a basket. You could call that unknown number $x$. Or say you earn $15 per hour and want to write your earnings without picking a specific number of hours: call the hours $h$, and your earnings are $15h$.

Variables let you represent unknown quantities, show relationships between values, and write general rules that work for many situations at once.

Examples of Variables in Action

Say you're saving for a new phone. You already have $50, and you plan to add $20 each week. How much will you have after some number of weeks? Let $w$ stand for the number of weeks. Your total savings are:

$$\text{Total savings} = 50 + 20w$$

After 3 weeks, substitute $3$ for $w$:

$$50 + 20 \cdot 3 = 50 + 60 = 110$$

You'd have $110 after 3 weeks.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols (like +, -, ×, ÷). Unlike an equation, an expression doesn't have an equals sign — it's just a phrase, not a complete sentence.

Here are some examples of algebraic expressions:

$x + 5$
$3y - 7$
$2a + 4b$
$\frac{m}{2} + 10$

Each of these expressions has some value, but that value depends on what number the variable represents. If $x = 3$, then $x + 5 = 8$. If $x = 10$, then $x + 5 = 15$.

Evaluating Expressions

Evaluating an expression means finding its value once you know what number the variable represents. Substitute the number for the variable, then calculate.

Example 1: Evaluate $2x + 3$ when $x = 4$

Substitute 4 for $x$: $$2(4) + 3 = 8 + 3 = 11$$

Example 2: Evaluate $5y - 12$ when $y = 6$

Substitute 6 for $y$: $$5(6) - 12 = 30 - 12 = 18$$

Example 3: Evaluate $3a + 2b$ when $a = 5$ and $b = 7$

Substitute both values: $$3(5) + 2(7) = 15 + 14 = 29$$

Understanding Equations

An equation is a mathematical statement that two expressions are equal. If an expression is a phrase, an equation is a complete sentence: it always has an equals sign (=) separating the two sides. Think of an equation as a balance scale — whatever is on the left has to match whatever is on the right.

Here are some simple equations:

$x + 3 = 7$
$2y = 10$
$4a - 5 = 11$

You may not know what $x$ is right away, but you do know that $3$ more than $x$ must equal $7$. What do you think $x$ is? Show answer$x = 4$. Three more than four is seven. The goal when working with equations is usually to solve them: find the value of the variable that makes the equation true.

Checking Solutions

Once you think you've found a solution to an equation, you can check your work by substituting your answer back into the original equation to see if both sides are equal.

Example: Is $x = 4$ a solution to the equation $x + 3 = 7$?

Substitute 4 for $x$:

$4 + 3 = 7$

$7 = 7$ ✓

Both sides equal 7, so $x = 4$ is the correct solution.

Example: Is $y = 3$ a solution to the equation $2y + 1 = 10$?

Substitute 3 for $y$:

$2(3) + 1 = 10$

$6 + 1 = 10$

$7 = 10$ ✗

No, this doesn't work. The two sides aren't equal, so $y = 3$ is not a solution.

The Language of Algebra

Algebra has its own vocabulary for describing operations. Translating between everyday English and algebraic expressions is one of the most important skills to build early on.

Here's how the basic operations show up in words:

Addition (+):

"the sum of $x$ and 5" → $x + 5$
"3 more than $y$" → $y + 3$
"increased by 7" → $n + 7$

Subtraction (-):

"the difference of $x$ and 4" → $x - 4$
"5 less than $y$" → $y - 5$
"decreased by 2" → $n - 2$

Multiplication (×):

"the product of 3 and $x$" → $3x$
"twice $y$" → $2y$
"5 times $a$" → $5a$

Note: In algebra, multiplication is usually written without the × symbol. You'll see $3x$ instead of $3 \times x$.

Division (÷):

"the quotient of $x$ and 2" → $\frac{x}{2}$
"$y$ divided by 4" → $\frac{y}{4}$
"half of $n$" → $\frac{n}{2}$

Translation Practice

Example 1: Write "7 more than twice a number" as an algebraic expression.

Let $n$ represent the unknown number. Think about it: what operation does "twice" imply, and how would you add 7 to that result? Show answer$2n + 7$. "Twice a number" is $2n$, and "7 more than" adds 7 to it.

Example 2: Write "the product of 5 and a number, decreased by 3" as an expression.

Let $x$ represent the number.

"product of 5 and a number" = $5x$
"decreased by 3" = subtract 3

Answer: $5x - 3$

Example 3: A rectangle has a width of 4 inches. If the length is $L$ inches, write an expression for the perimeter.

The perimeter of a rectangle is $2 \times \text{length} + 2 \times \text{width}$, so:

$$2L + 2(4) = 2L + 8$$

Practice Problems

Try these on your own, then check your answers below.

Evaluate $4x - 7$ when $x = 5$ Show answer$4(5) - 7 = 20 - 7 = 13$
Evaluate $3a + 2b$ when $a = 4$ and $b = 6$ Show answer$3(4) + 2(6) = 12 + 12 = 24$
Is $x = 8$ a solution to $x - 3 = 5$? Show answerSubstitute: $8 - 3 = 5$, which gives $5 = 5$. Yes, $x = 8$ is a solution.
Is $y = 2$ a solution to $3y + 4 = 11$? Show answerSubstitute: $3(2) + 4 = 6 + 4 = 10$, which gives $10 = 11$. No, $y = 2$ is not a solution.
Write "6 less than 3 times a number" as an algebraic expression. Show answerLet $n$ be the number: $3n - 6$
Write "the sum of a number and 9, divided by 2" as an algebraic expression. Show answerLet $x$ be the number: $\frac{x + 9}{2}$

What's Next?

Two small pitfalls to watch out for as you keep going. First, "less than" can trip you up: "5 less than $x$" means $x - 5$, not $5 - x$ — the number being subtracted from comes first. Second, when you solve an equation, always substitute your answer back into the original to check that it works.

From here, the next stops are order of operations — the rules that tell you which calculation to do first in an expression — and properties of numbers, which describe the moves you're allowed to make when simplifying. After that, you'll move on to actually solving equations for an unknown.

Algebra feels different from arithmetic at first. The more you practice translating between words and symbols, the more natural it gets.