Algebra Refresher for Problem Solvers
Variables and Expressions
The Language of Algebra
At its heart, algebra is about finding unknown values. To do that, we need a way to talk about numbers we don't know yet. We use symbols as placeholders for these mystery numbers.
Variable
noun
A symbol, usually a letter, that represents an unknown number or a quantity that can change.
Imagine you're buying some avocados. You know you want 4 of them, but you don't know the price of a single avocado. You can use a variable, like , to stand for the unknown price.
The total cost for the avocados would be 4 times the price of one, which we can write as , or more simply, . When a number is right next to a variable, it means multiplication.
This combination of numbers, variables, and operations ($4a$) is called an algebraic expression. It's a mathematical phrase, not a full sentence. It doesn't have an equals sign.
Let's expand on that. If you buy 4 avocados and a bag of chips that costs $3, your total cost is represented by the expression $4a + 3$. This expression has different parts:
- Terms: The parts of the expression separated by addition or subtraction signs. In $4a + 3$, the terms are $4a$ and $3$.
- Coefficient: The number multiplied by a variable. In the term $4a$, the coefficient is $4$.
- Constant: A term without a variable. Its value never changes. Here, the constant is $3$.
| Expression | Terms | Coefficient(s) | Constant(s) | Variable(s) |
|---|---|---|---|---|
7x - 2 | 7x, -2 | 7 | -2 | x |
5a + b + 9 | 5a, b, 9 | 5, 1 | 9 | a, b |
y/3 | y/3 | 1/3 | none | y |
Simplifying Expressions
Often, algebraic expressions can look messy. Simplifying them makes them easier to work with. Two key techniques for this are combining like terms and using the distributive property.
Like terms are terms that have the same variable raised to the same power. For example, $3x$ and $5x$ are like terms. $3x$ and $5x^2$ are not.
Combining like terms is like organizing groceries. You put all the apples together and all the oranges together. You can add or subtract the coefficients of like terms.
Consider the expression $2x + 5 + 3x$. The terms $2x$ and $3x$ are like terms. We can combine them:
$2x + 3x = (2+3)x = 5x$
So, the simplified expression is $5x + 5$.
Another powerful tool is the distributive property. It lets you multiply a sum by multiplying each addend separately and then adding the products.
Let's see it in action with the expression .
We distribute the to both the and the :
Putting it together, simplifies to . We can't simplify this any further because and are not like terms.
You buy 'p' pencils and an eraser that costs 2, which expression represents the total cost?
In the expression , what is the constant term?
Understanding variables and how to simplify expressions is the first major step into algebra. These skills are the foundation for solving equations and much more.