2.1 Use the Language of Algebra

Algebraic expressions and equations are the building blocks of algebra. They use variables and symbols to represent unknown quantities and relationships, letting you model real-world situations and solve problems mathematically.

Understanding the components of expressions (like coefficients and constants) is crucial. Knowing how to simplify expressions using exponent rules and the order of operations gives you the tools to manipulate and evaluate these expressions correctly.

Algebraic Expressions and Equations

Variables and algebraic symbols

A variable is a letter that stands in for an unknown quantity. The most common ones you'll see are $x$, $y$, $a$, and $b$, but any letter can be used.

Algebraic symbols tell you what operation to perform:

• Addition: the plus sign $+$
• Subtraction: the minus sign $-$
• Multiplication: the times sign $\times$, a dot $\cdot$, or simply placing things next to each other like $(ab)$ or $3x$
• Division: the division sign $\div$ or fraction notation $\frac{a}{b}$
• Equality: the equal sign $=$

When you combine variables, numbers, and operation symbols, you get either an algebraic expression (like $3x + 2$) or an equation (like $2x - 5 = 7$). The difference between those two is important, and that's covered next.

Expressions vs equations

The key distinction here is simple: expressions don't have an equal sign, equations do.

An algebraic expression combines variables, numbers, and operations but never states that two things are equal. Examples: $4x$, $2y - 3$, $\frac{x}{2} + 5$. You can simplify an expression (by combining like terms, for instance), but you don't "solve" it because there's nothing to solve for.

An algebraic equation uses an equal sign to say two expressions have the same value. Examples: $3x = 12$, $y - 2 = 5$, $\frac{x}{4} + 3 = 7$. Because an equation makes a claim about equality, you can solve it to find the value of the variable that makes it true.

A quick way to remember: if you see $=$, it's an equation. If you don't, it's an expression.

Components of Algebraic Expressions

Every algebraic expression is made up of terms separated by addition or subtraction. Each term has specific parts worth knowing:

• Coefficient: The number multiplied by the variable in a term. In $3x^2$, the coefficient is 3. If you see just $x$, the coefficient is 1 (it's implied).
• Constant: A term with a fixed value and no variable attached. In $2x + 5$, the constant is 5.
• Like terms: Terms that have the exact same variable(s) raised to the exact same power(s). For example, $3x$ and $5x$ are like terms because they both have $x$ to the first power. You can combine them: $3x + 5x = 8x$. But $3x$ and $3x^2$ are not like terms because the exponents differ.

Two other concepts you'll use often:

• Distributive property: This lets you multiply a number across a sum or difference inside parentheses. For example, $a(b + c) = ab + ac$. With numbers: $3(x + 4) = 3x + 12$.
• Algebraic fraction: A fraction where the numerator, denominator, or both contain algebraic expressions, like $\frac{x+1}{x-2}$.

Exponent rules for simplification

An exponent tells you how many times to multiply a base by itself. For example, $3^2 = 3 \times 3 = 9$.

Here are the core rules for working with exponents:

• Multiplying same bases — add the exponents: $2^3 \times 2^4 = 2^{3+4} = 2^7$
• Dividing same bases — subtract the exponents: $\frac{x^5}{x^2} = x^{5-2} = x^3$
• Power raised to a power — multiply the exponents: $(3^2)^4 = 3^{2 \times 4} = 3^8$
• Negative exponents — flip the base to a fraction: $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$

These rules only work when the bases are the same. You can't combine $2^3 \times 3^4$ using the addition rule because the bases (2 and 3) are different.

Order of operations in expressions

When an expression has multiple operations, you need to evaluate them in a specific order. The mnemonic PEMDAS helps you remember:

1. P — Parentheses: Simplify everything inside parentheses first.
2. E — Exponents: Evaluate exponents next.
3. MD — Multiplication and Division: Work these from left to right (they have equal priority).
4. AS — Addition and Subtraction: Work these from left to right (they also have equal priority).

A common mistake is thinking multiplication always comes before division, or addition before subtraction. That's not the case. Multiplication and division are done left to right, and addition and subtraction are done left to right.

Here's a worked example with $2 + 3 \times (4^2 - 1) \div 5$:

1. Parentheses first: Inside you have $4^2 - 1$. The exponent comes first: $4^2 = 16$. Then subtract: $16 - 1 = 15$.

2. Exponents: Already handled inside the parentheses.

3. Multiplication and Division (left to right): $3 \times 15 = 45$, then $45 \div 5 = 9$.

4. Addition: $2 + 9 = 11$.

Following the correct order of operations ensures you get the right answer every time, especially in multi-step problems.