1.2 Use the Language of Algebra

Algebraic expressions are the building blocks of algebra. They use variables, symbols, and numbers to represent mathematical relationships. Understanding how to read and work with these expressions is crucial for solving equations and modeling real-world scenarios.

Algebraic Expressions and Variables

Variables and algebraic symbols

A variable is a letter that stands for an unknown or changing quantity. You'll most often see $x$, $y$, or $z$, but any letter can be used.

Algebraic symbols represent the operations connecting those quantities:

• Addition (+), subtraction (-)
• Multiplication ($\cdot$ or $\times$ or parentheses, as in $2(3)$)
• Division ($\div$ or fraction notation)
• Equality (=)

A constant is a fixed value that doesn't change. In the expression $3x + 5$, the number 5 is a constant, and 3 is the numerical coefficient (the number multiplying the variable $x$).

Order of operations in expressions

When an expression has more than one operation, you need a consistent rule for which to do first. That rule is PEMDAS:

1. Parentheses: simplify anything inside grouping symbols first
2. Exponents: evaluate powers and roots next
3. Multiplication and Division: work left to right
4. Addition and Subtraction: work left to right

Multiplication and division share the same priority level, and so do addition and subtraction. Within each level, you simply move left to right.

Example: Simplify $3 + 2 \cdot (4 - 1)^2$

1. Parentheses: $4 - 1 = 3$

2. Exponents: $3^2 = 9$

3. Multiplication: $2 \cdot 9 = 18$

4. Addition: $3 + 18 = 21$

A common mistake is to add $3 + 2$ first and get $5 \cdot 9 = 45$. That's wrong because multiplication comes before addition.

Substitution for expression evaluation

Substitution means replacing a variable with a given number and then simplifying. Whenever you substitute, it helps to put the number inside parentheses so you don't lose track of signs or operations.

Example: Evaluate $2x - 3$ when $x = 5$

1. Replace $x$ with 5: $2(5) - 3$

2. Multiply: $10 - 3$

3. Subtract: $7$

If the expression has more than one variable, replace each one with its given value and then follow the order of operations as usual.

Like terms in expressions

Like terms have the same variable(s) raised to the same exponent(s). For example, $3x$ and $5x$ are like terms because they both contain $x^1$. But $3x$ and $5x^2$ are not like terms because the exponents differ.

To combine like terms, add or subtract their coefficients:

$3x + 5x = 8x$

The distributive property lets you multiply a factor across terms inside parentheses:

$2(3x + 4) = 6x + 8$

You multiply the 2 by each term separately: $2 \cdot 3x = 6x$ and $2 \cdot 4 = 8$.

Verbal to algebraic conversion

A big part of algebra is turning word problems into expressions. Here's how:

1. Identify the unknown quantity and assign it a variable (say, $x$).

2. Translate key words into operations:

   • "sum" or "plus" → addition (+)
   • "difference" or "minus" → subtraction (-)
   • "product" or "times" → multiplication ($\cdot$)
   • "quotient" or "divided by" → division ($\div$ or a fraction)

3. Build the expression following the structure of the sentence.

Example: "The sum of three times a number and five"

• "a number" → $x$
• "three times a number" → $3x$
• "the sum of ... and five" → $3x + 5$

Watch the word order carefully. "Five less than a number" is $x - 5$, not $5 - x$. The phrase "less than" flips the order from how you read it in English.

Algebraic notation and simplification

Simplifying an expression means reducing it to its most concise form. You do this by combining like terms and carrying out any operations you can.

For example, $4x + 3 + 2x - 1$ simplifies to $6x + 2$ because you combine $4x + 2x = 6x$ and $3 - 1 = 2$.

A fully simplified expression has no like terms left to combine and no parentheses left to distribute. Building comfort with simplification now will make solving equations much smoother later on.