Introduction to Algebra

Algebra is a branch of mathematics where we use letters and symbols to represent numbers, often because we don't know what the number is right away or because we want to explore how one number is affected by another. We use formulas and equations to explain how these letters and symbols (called variables and constants) are related. If you've been working with arithmetic (regular math with just numbers), algebra is the next step. Instead of always working with specific numbers, we'll use variables to represent unknown values or quantities that can change.

Think of algebra as a powerful tool that helps us solve real-world problems. Whether you're figuring out how much money you'll save over time, calculating distances for a road trip, or even determining the right amount of ingredients for a recipe, algebra gives us a way to work with unknowns and find answers.

What is a Variable?

A variable is a letter or symbol that stands in for a number we don't know yet, or a number that can change. We typically use letters like $x$, $y$, or $z$ as variables, though any letter works as long as you understand what it represents.

For example, if we don't know how many apples are in a basket, we might call that unknown number $x$. Or, if you earn $15 per hour but don't know how many hours you'll work, we might call the hours $h$ and calculate your earnings as 15 times $h$, which we'd write as simple $15h$.

Variables are incredibly useful because they let us represent unknown quantities, show relationships between different values, and rrite general rules that work for many different situations.

Examples of Variables in Action

Let's say you're saving money for a new phone. You already have $50, and you plan to save $20 each week. How can we calculate the amount of money you have after a certain number of weeks? Well, we could use $w$ to represent the number of weeks, and write your total savings as:

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

If you save for 3 weeks, just substitute $3$ instead of that $w$: $$50 + 20*3$$ $$50 + 60$$ $$110$$

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 when you know what number the variable represents. You simply substitute the number for the variable and 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 shows two expressions are equal. Like we said before, if an expression is a phrase, then an equation is a complete sentence. An equation always has an equals sign (=) separating the two sides. Think of an equation like a balance scale—whatever is on the left side must equal whatever is on the right side.

Here are some simple equations:

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

We don't necessarily know what $x$ is right away, but we know $3$ more than x must equal $7$. What do you think $x$ might be? The goal when working with equations is usually to solve them, which means finding the value of the variable that makes the equation true. In this case, $x=4$. Three more than $x$ is $7$ only if $x$ is four.

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 $x$:

$4 + 3 = 7$

$7 = 7$ ✓

Yes! Since both sides equal 7, we know $x = 4$ 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 mathematical operations. Learning to translate between everyday English and algebraic expressions is a key skill.

Here's examples of how we might express the basic operations you already know well:

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, we usually write multiplication without the × symbol. We write $3x$ 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.

"twice a number" = $2n$
"7 more than" = add 7

Answer: $2n + 7$

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.

Remember, perimeter = $2 \times \text{length} + 2 \times \text{width}$

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

Practice Problems

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

Evaluate $4x - 7$ when $x = 5$
Evaluate $3a + 2b$ when $a = 4$ and $b = 6$
Is $x = 8$ a solution to $x - 3 = 5$?
Is $y = 2$ a solution to $3y + 4 = 11$?
Write "6 less than 3 times a number" as an algebraic expression.
Write "the sum of a number and 9, divided by 2" as an algebraic expression.

Solutions

$4(5) - 7 = 20 - 7 = 13$
$3(4) + 2(6) = 12 + 12 = 24$
Substitute: $8 - 3 = 5$, which gives us $5 = 5$. Yes, $x = 8$ is a solution.
Substitute: $3(2) + 4 = 6 + 4 = 10$, which gives us $10 = 11$. No, $y = 2$ is not a solution.
Let $n$ be the number: $3n - 6$
Let $x$ be the number: $\frac{x + 9}{2}$

Common Mistakes to Avoid

Forgetting the order of operations: When evaluating $2x + 3$ with $x = 4$, you must multiply first: $2(4) + 3 = 8 + 3 = 11$, not $2(4 + 3)$.

Confusing "less than" with subtraction order: "5 less than $x$" means $x - 5$, not $5 - x$. The number being subtracted FROM comes first. Think it through carefully, and make sure the math you write expresses precisely what you meant.

Not checking your work: Always substitute your answer back into the original equation to verify it works.

What's Next?

Now that you understand variables, expressions, and equations, you're ready to start working with them. In the next lessons, we'll learn about the order of operations (making sure we calculate expressions correctly), properties of numbers (rules that make algebra easier), and eventually how to solve equations to find unknown values.

Remember: algebra might feel different from regular math at first, but it's really just a tool for solving problems. The more you practice translating between words and algebra, the more natural it will become.