Lesson 2: Understanding Variables and Expressions

New Concept

Welcome to Algebra! At its heart is the concept of a variable: a symbol, usually a letter, used to represent an unknown number.

What’s next

Now, let's start decoding this new language. We’ll break down expressions and practice identifying their essential parts: variables, constants, coefficients, and terms.

Property

A symbol, usually a letter, used to represent an unknown number is called a variable. A quantity whose value does not change is called a constant.

Examples

In the expression $9 + b$, the constant is $9$ and the variable is $b$.
For $25pq - 11$, the constants are $25$ and $11$, while the variables are $p$ and $q$.
The expression for a movie ticket cost of $12.50$ dollars per person could be $12.50p$.

Explanation

Think of a variable like a mystery box in a video game—its value can change depending on the situation! Constants are the opposite; they're like the price of an item in a shop, always staying the same. In algebra, we use letters like $x$ for these mystery numbers and regular numbers for the constants that stay put.

Property

The numeric factor of a product including a variable is called the numeric coefficient, or simply the coefficient.

Examples

In the term $14ab$, the coefficient is $14$.
For the term $\frac{z}{4}$, the coefficient is $\frac{1}{4}$ because it's the same as $\frac{1}{4}z$.
The term $uvw$ has an implied coefficient of $1$.

Explanation

A coefficient is the number that's best friends with a variable, telling you how many of that variable you have. If you see $8y$, the coefficient is $8$. It's the numerical part being multiplied by the letter part. Watch out for invisible coefficients! If you just see a variable like $x$, its coefficient is a secret agent '1'.

Property

Parts of an expression separated by $+$ or $-$ signs are called terms of an expression.

Examples

The expression $7x + 2y - 15$ has three terms: $7x$, $2y$, and $-15$.
The expression $11ab - (c+3)$ has two terms: $11ab$ and $-(c+3)$.
In the expression $4xy + \frac{9z}{2+w} - 1$, there are three terms.

Explanation

Think of an algebraic expression as a train. Each car, separated by a plus or minus sign, is a 'term.' These terms are the individual chunks that make up the whole mathematical sentence. Even something complex inside parentheses like $(x+5)$ can be a single term if it's being treated as one block in the bigger expression.