Foundations of Algebra

Many people found themselves enjoying math right up until they encountered algebra. So, if algebra seems scary to you, know that you are not alone. But here is some good news: you already do algebra every single day without realizing it. When you figure out how much money you need to save each week to afford something, or when you calculate how long a trip will take at a certain speed, you are thinking algebraically. We are just going to give those thought processes a proper vocabulary and some tools to write them down.

Think about it this way: algebra is not some mystical system invented to torment students. It is simply a language - a way to ask and answer questions about numbers when you do not know all of them yet. When you see $3 + ? = 7$ and think “the answer is 4,” you have just solved an algebra problem. The only difference is that in algebra, we use letters like $x$ instead of question marks. Same idea, slightly different notation.

Core Concepts

What is Algebra?

Algebra is often called the “language of mathematics” - and that is a helpful way to think about it. Just as English gives you words to express ideas, algebra gives you symbols and rules to express mathematical relationships. It lets you describe patterns, solve problems, and answer questions like “how much?” or “how many?” even when you do not have all the information yet.

At its heart, algebra is about one thing: asking questions. Every algebraic equation is really just asking, “What value makes this statement true?” When you see $x + 5 = 12$, the equation is asking, “What number, when you add 5 to it, gives you 12?” You already know the answer is 7. Algebra just gives you a systematic way to find and express such answers.

Variables: Placeholders for the Unknown

A variable is a letter (or symbol) that stands in for a number you do not know yet, or a number that can change. That is the whole mystery demystified. Variables are just placeholders.

Why do we use letters instead of just leaving blanks or using question marks? Because letters let us:

• Talk about unknown values before we figure them out
• Write formulas that work for many different situations
• Describe how quantities relate to each other

Common variables you will see: $x$, $y$, $n$, $t$, $a$, $b$

The choice of letter is often arbitrary, though sometimes we pick meaningful ones (like $t$ for time or $d$ for distance). The letter itself does not change how the math works.

Constants vs. Variables

A constant is the opposite of a variable - it is a value that stays fixed and does not change. The number 5 is always 5. The number $\pi$ is always approximately 3.14159. These are constants.

In the expression $3x + 7$:

• The $x$ is a variable (it could represent different numbers in different situations)
• The 3 and 7 are constants (they always mean exactly 3 and exactly 7)
• The 3 is also called a coefficient because it is multiplied by the variable

Understanding the difference matters because when you work with an expression, the constants stay put while the variables are what you substitute values into or solve for.

Algebraic Expressions vs. Equations

These two terms are often confused, but the distinction is simple:

An expression is a mathematical phrase - it contains numbers, variables, and operations, but it does not make a claim. It is like a noun phrase: “three times a number plus five” or $3x + 5$. You can evaluate it, simplify it, but you cannot “solve” it because it is not claiming anything is equal to anything else.

An equation is a complete mathematical sentence. It states that two expressions are equal. It is like a declaration: “three times a number plus five equals twenty” or $3x + 5 = 20$. Now you have something to solve - you can find the value of $x$ that makes this statement true.

The key visual difference: equations have an equals sign (=), expressions do not.

Order of Operations Review (PEMDAS)

When evaluating algebraic expressions, you need to perform operations in the correct order. The standard order is:

1. Parentheses (do what is inside first)
2. Exponents (powers and roots)
3. Multiplication and Division (left to right)
4. Addition and Subtraction (left to right)

This order matters. Consider $2 + 3 \times 4$. If you went left to right, you would get $(2 + 3) \times 4 = 20$. But following PEMDAS, you do multiplication first: $2 + (3 \times 4) = 2 + 12 = 14$. The correct answer is 14.

With variables, the same rules apply. To evaluate $3x^2 + 2x$ when $x = 4$:

1. Substitute: $3(4)^2 + 2(4)$
2. Exponents first: $3(16) + 2(4)$
3. Multiplication: $48 + 8$
4. Addition: $56$

Evaluating Expressions by Substitution

Evaluating an expression means finding its numerical value when you replace each variable with a specific number. Think of it as filling in the blanks.

The process is straightforward:

1. Write the expression
2. Replace each variable with the given value (use parentheses to be safe)
3. Follow the order of operations to calculate

For example, to evaluate $5x - 3$ when $x = 7$:

• Substitute: $5(7) - 3$
• Multiply: $35 - 3$
• Subtract: $32$

Using parentheses when you substitute helps prevent errors, especially with negative numbers. If $x = -2$, writing $5(-2) - 3$ is much clearer than $5 \cdot -2 - 3$.

Translating Words into Algebraic Expressions

One of the most practical algebra skills is converting everyday language into mathematical expressions. Here is a guide to common phrases:

Words	Operation	Example Phrase	Expression
sum, plus, more than, increased by, total	Addition	eight more than a number	$x + 8$
difference, minus, less than, decreased by	Subtraction	a number decreased by 4	$x - 4$
product, times, of, multiplied by	Multiplication	twice a number	$2x$
quotient, divided by, per, ratio	Division	a number divided by 5	$\frac{x}{5}$

Watch out for “less than”! The phrase “seven less than a number” means $x - 7$, not $7 - x$. You start with the number and take away 7. This trips up many students, so pay attention to the order.

Notation and Terminology

Term	Meaning	Example
Variable	A letter representing an unknown quantity	$x$, $y$, $n$
Constant	A fixed value that does not change	5, $\pi$, -3
Expression	A mathematical phrase with numbers, variables, operations	$3x + 7$
Equation	A statement that two expressions are equal	$3x + 7 = 16$
Term	A part of an expression separated by + or -	In $3x + 7$, terms are $3x$ and $7$
Coefficient	The number multiplied by a variable	In $3x$, the coefficient is 3

A note about multiplication notation: In algebra, we usually skip the multiplication sign because it looks too much like the variable $x$. Instead of $3 \times x$, we write $3x$. Instead of $a \times b$, we write $ab$ or $a \cdot b$. This is just a convention to keep things readable.

Examples

Key Properties and Rules

The Structure of Algebraic Expressions

Every algebraic expression is built from terms. Terms are the parts separated by addition and subtraction signs. In the expression $4x^2 - 3x + 7$:

• $4x^2$ is the first term
• $-3x$ is the second term (keep the negative sign with it)
• $7$ is the third term (a constant term)

Each term with a variable has a coefficient - the number multiplied by the variable part:

• In $4x^2$, the coefficient is 4
• In $-3x$, the coefficient is $-3$
• In just $x$, the coefficient is 1 (since $x = 1 \cdot x$)

Order of Operations in Algebra

When working with algebraic expressions, always follow PEMDAS:

1. Parentheses/Brackets - simplify inside first
2. Exponents - evaluate powers
3. Multiplication and Division - left to right
4. Addition and Subtraction - left to right

This order ensures everyone gets the same answer from the same expression. It is the grammar of mathematical language.

Substitution Guidelines

When substituting values into expressions:

1. Always use parentheses around the substituted value
2. Be especially careful with negative numbers
3. Follow the order of operations after substituting
4. Double-check your signs at each step

Example: For $x^2 - 4x$ when $x = -3$:

• Write: $(-3)^2 - 4(-3)$
• Not: $-3^2 - 4 \cdot -3$ (this is ambiguous and error-prone)

Real-World Applications

Phone and Subscription Pricing

Many services charge a base fee plus a variable amount. If a streaming service costs $10/month plus $3 per premium channel, and you have $c$ premium channels, your bill is $10 + 3c$. This formula works whether you have 0 channels or 20.

Distance, Rate, and Time

The relationship $d = rt$ (distance equals rate times time) is one of the most useful algebraic formulas. If you drive at 60 mph for $t$ hours, you travel $60t$ miles. Planning a 300-mile trip? Set up $300 = 60t$ and solve to find you need 5 hours.

Temperature Conversion

Converting Celsius to Fahrenheit uses the formula $F = \frac{9}{5}C + 32$. This expression takes a temperature in Celsius and outputs the equivalent in Fahrenheit. At $C = 20$ (a pleasant day), you get $F = \frac{9}{5}(20) + 32 = 36 + 32 = 68°F$.

Tips and Discounts

To calculate a 20% tip on a meal costing $m$ dollars, use $0.20m$ (or equivalently, $\frac{m}{5}$). For the total bill with tip: $m + 0.20m = 1.20m$. A $45 meal with 20% tip costs $1.20(45) = $54$.

For a 25% discount on an item priced at $p$ dollars: the discount is $0.25p$, so you pay $p - 0.25p = 0.75p$. A $80 item at 25% off costs $0.75(80) = $60$.

Self-Test Problems

Summary

• Algebra is the language of mathematics - a way to express relationships and ask questions about unknown quantities using symbols and rules.
• A variable is a letter representing an unknown or changing value. It is simply a placeholder until you know (or assign) its actual value.
• A constant is a fixed value that does not change, like 5 or $\pi$.
• An expression is a mathematical phrase combining numbers, variables, and operations (no equals sign). An equation states that two expressions are equal (has an equals sign).
• Terms are the parts of an expression separated by + and - signs. Coefficients are the numbers multiplied by variables.
• To evaluate an expression, substitute the given value(s) for the variable(s) and calculate, following the order of operations (PEMDAS).
• When translating words to algebra, identify key phrases: “more than” means add, “less than” means subtract (carefully), “product” means multiply, “quotient” means divide.
• Algebraic expressions appear everywhere in real life: pricing formulas, distance calculations, temperature conversions, tips, and discounts. Learning to write and evaluate them gives you a powerful tool for solving everyday problems.