Lesson 1.2: Use the Language of Algebra

New Concept

This lesson introduces the language of algebra. You'll learn how to use variables and symbols to build expressions, evaluate them for specific values, and simplify them by following the order of operations and combining like terms.

What’s next

Next, you’ll apply these rules through interactive examples on simplifying expressions and then tackle challenges that involve translating words into algebraic language.

Property

A variable is a letter that represents a number whose value may change.
A constant is a number whose value always stays the same.

Examples

• In the formula for the area of a square, $A = s^2$, the area $A$ and side length $s$ are variables, as they can change depending on the square.
• If a taxi charges 3 dollars per mile, the cost per mile is a constant. The total fare and the distance traveled are variables.
• The number of days in January, 31, is a constant. The number of sunny days in January, $d$, is a variable.

Explanation

Variables are placeholders for values that can change, like your age or the temperature. Constants are fixed numbers that do not change, like the number of hours in a day. We use them together to build algebraic relationships.

Property

An expression is a number, a variable, or a combination of numbers and variables using operation symbols.
An equation is two expressions connected by an equal sign.

Examples

• The phrase 'the sum of a number $x$ and 5' translates to the expression $x + 5$. There is no equal sign.
• The sentence '$y$ plus nine is equal to two $y$ minus three' is an equation written as $y + 9 = 2y - 3$.
• $4(z-1) + 5$ is an expression because it represents a value but does not make a complete statement with an equal sign.

Explanation

Think of an expression as a mathematical phrase, like 'a number minus three'. An equation is a complete mathematical sentence that states two expressions are equal, like 'a number minus three equals ten'.

Property

$a^n$ means multiply $a$ by itself, $n$ times.

The expression $a^n$ is read as $a$ to the $n$th power. In this notation, $a$ is the base and $n$ is the exponent.

Examples

• The expression $4^3$ means $4 \cdot 4 \cdot 4$, which simplifies to $64$. Here, 4 is the base and 3 is the exponent.
• The product $y \cdot y \cdot y \cdot y \cdot y$ can be written in exponential notation as $y^5$.
• When evaluating $2^x$ for $x=4$, we substitute to get $2^4$, which means $2 \cdot 2 \cdot 2 \cdot 2 = 16$.

Explanation

Exponential notation is a powerful shorthand for writing repeated multiplication. Instead of writing a long string of numbers, you can use a base and an exponent to represent the same value much more concisely.

Property

Step 1. Parentheses and Other Grouping Symbols
Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first.
Step 2. Exponents
Simplify all expressions with exponents.
Step 3. Multiplication and Division
Perform all multiplication and division in order from left to right. These operations have equal priority.
Step 4. Addition and Subtraction
Perform all addition and subtraction in order from left to right. These operations have equal priority.

Examples

• To simplify $30 - 5 \cdot 4$, we perform multiplication first: $30 - 20 = 10$.
• In the expression $(3+2)^2 \div 5$, we start with parentheses $(5)^2 \div 5$, then the exponent $25 \div 5$, and finally division to get $5$.
• For $4 + 2[10 - 3(2)]$, we work inside the innermost parentheses first: $4 + 2[10 - 6]$, then inside the brackets $4 + 2[4]$, then multiply $4 + 8$, and finally add to get $12$.

Explanation

The order of operations (PEMDAS) is a set of rules everyone follows to solve math problems. This ensures that every expression has only one correct answer, preventing confusion and making sure our calculations are consistent.

Property

To combine like terms, first identify the terms that have the same variables and exponents. Next, rearrange the expression so the like terms are grouped together. Finally, add or subtract the coefficients of the like terms to simplify the expression.

Examples

• To simplify $7a + 4b + 2a$, we identify $7a$ and $2a$ as like terms and combine them to get $9a + 4b$.
• In the expression $5x^2 + 9 + 2x^2 - 4$, we combine the $x^2$ terms to get $7x^2$ and the constants to get $5$. The result is $7x^2 + 5$.
• To simplify $10y + 3y^2 + 2y + 6y^2$, we combine the $y^2$ terms to get $9y^2$ and the $y$ terms to get $12y$. The simplified expression is $9y^2 + 12y$.

Explanation

Combining like terms simplifies an expression by grouping similar items. Just as you would group 3 apples and 4 apples to get 7 apples, you combine terms like $3x$ and $4x$ to get $7x$.

Property

To translate an English phrase into an algebraic expression, first identify keywords that indicate the mathematical operation. Words like 'sum' or 'more than' mean addition, 'difference' or 'less than' mean subtraction, 'product' means multiplication, and 'quotient' means division. Then, represent unknown numbers with variables and combine them according to the phrase.

Examples

• The phrase 'ten less than a number $n$' translates to the expression $n - 10$.
• 'The product of 4 and the sum of $x$ and $y$' requires parentheses for the sum, resulting in the expression $4(x+y)$.
• 'The number of dimes is five more than twice the number of nickels, $n$' translates to the expression $2n + 5$.

Explanation

Translating from English to algebra is like being a language interpreter. You convert everyday words into the precise language of mathematics, which allows you to set up and solve real-world problems systematically.