Welcome back to our math series! Today, we are diving into the foundational world of Algebra. If you’ve ever looked at a math problem and wondered why there are letters mixed in with the numbers, this post is for you. Algebra is essentially a language used to describe patterns and relationships in mathematics.
Let’s break down the basic building blocks.
In algebra, we often deal with values that aren’t fixed. These are called Variables.
Definition: A symbol (usually a letter) that can take on various numerical values is called a variable or a literal.
Examples: Common variables include letters like $x$, $y$, $z$, $a$, $b$, and $c$.
Think of a variable as a placeholder or a container. Depending on the problem, that container could hold any number!
While variables can change, some values stay the same. These are called Constants.
Definition: A symbol that has a fixed, unchanging value is called a constant.
Examples: Numbers like $5$, $-3$, and $2$ are all constants because their value never changes.
One of the most important skills in algebra is “translating” English sentences into mathematical expressions (symbolic form). Here are some common examples of how to do this:
| Word Statement | Symbolic Form |
| The sum of $a$ and $b$ | $a + b$ |
| The subtraction of $b$ from $a$ | $a – b$ |
| The product of $2$ and $x$ | $2 \times x$ or $2x$ |
| Divide $x$ by $2$ | $\frac{x}{2}$ |
| $6$ more than $x$ | $x + 6$ |
| $5$ decreased by $x$ | $5 – x$ |
| $5$ times $x$ | $5x$ |
| $40$ divided by $x$ | $\frac{40}{x}$ |
| $5$ times $x$ added to $3$ times $y$ | $5x + 3y$ |
| The subtraction of $2x$ from $3$ | $3 – 2x$ |
| $4$ less than half of $x$ | $\frac{x}{2} – 4$ |
An algebraic term is the basic unit of an expression. It can be:
A constant alone: For example, $6$.
A variable alone: For example, $x$.
A combination of both: This happens through the operations of multiplication or division.
Examples: $6x$ and $\frac{6}{x}$ are both considered terms.
When you have a product (the result of multiplication), the individual parts that make up that product are called factors.
Definition: In a product, each literal (variable) or numerical value is a factor of that product.
Example: In the product $5xy$, which is $5 \times x \times y$, the values $5$, $x$, and $y$ are all called factors of $5xy$.
A co-efficient describes the relationship between factors within a single product.
Definition: In a product containing two or more factors, each factor is called the co-efficient of the product of the other factors.
Common Examples:
In the term $5x$, $5$ is the co-efficient of $x$.
In the term $-6xy$, $-6$ is the co-efficient of $xy$.