Lesson 2.4: Algebraic Expressions

New Concept

An algebraic expression combines numbers and variables using operations like addition or multiplication. This lesson teaches you how to translate real-world situations into this mathematical language and evaluate expressions to find specific numerical answers.

What’s next

Now, let's break down how to build these expressions. You'll work through interactive examples for sums, products, differences, and quotients.

Property

When we add two numbers or variables together, the result is called the sum, and the things added together are called terms. When we multiply two numbers or variables together, the result is called the product, and the things multiplied together are called factors. In algebra, we write the product of a number and a variable next to each other without any symbol, for example, $5g$ means $5 \times g$. Addition and multiplication are commutative, meaning the order of the terms or factors does not change the result.

Examples

• The sum of a variable $x$ and 10 is written as $x + 10$.
• The product of 4 and a variable $n$ is written as $4n$.
• The product of the variables $a$ and $b$ is written as $ab$.

Explanation

A 'sum' is the result of adding terms, while a 'product' is the result of multiplying factors. In algebra, we often write multiplication by placing a number and variable side-by-side, like $3x$, to make it simpler.

Property

When we subtract one number or variable from another, the result is called the difference. For example, '$b$ subtracted from $12$' means $12-b$. When we divide one number or variable by another, the result is called the quotient. We often use a fraction bar for division, so $\frac{20}{5}$ means $20 \div 5$. The ratio of $a$ to $b$ is the quotient $\frac{a}{b}$. Unlike addition and multiplication, subtraction and division are not commutative, so the order of the numbers matters.

Examples

• The phrase '7 subtracted from $y$' is written as the expression $y - 7$.
• The quotient of a variable $P$ and 4 is written using a fraction bar as $\frac{P}{4}$.
• The ratio of your points $p$ to a total of 50 is written as $\frac{p}{50}$.

Explanation

A 'difference' is the answer to a subtraction problem, and a 'quotient' is the answer to a division problem. Be careful! The order is very important for both of these operations. $8 - 2$ is different from $2 - 8$.

Property

If we know the value of the variable, we can substitute this value into the expression and simplify the result. This process is called evaluating the algebraic expression.

Examples

• To evaluate $a + 8$ for $a = 5$, we substitute to get $5 + 8 = 13$.
• To evaluate $7x$ for $x = 3$, we substitute to get $7(3) = 21$.
• To evaluate $\frac{n}{4}$ for $n = 28$, we substitute to get $\frac{28}{4} = 7$.

Explanation

'Evaluating' means to plug in a given number for a variable and find the final value. You are simply replacing the letter with its known number and then doing the math to solve it. It turns algebra back into arithmetic.

Property

To write an algebraic expression from an English phrase, follow these steps:

1. Identify the unknown quantity.
2. Choose a variable to represent the unknown quantity.
3. Translate the English phrase into an algebraic expression, using the variable and appropriate operation symbols. Phrases with 'of,' such as 'three-quarters of' or 'eight percent of,' indicate multiplication.

Examples

• The phrase '10 dollars more than the price $p$' translates to the expression $p + 10$.
• The phrase 'the cost $c$ split 5 ways' translates to the expression $\frac{c}{5}$.
• The phrase '15% of the total $T$' translates to the expression $0.15T$.

Explanation

Think of this as translating from English into the language of math. Identify the unknown value, give it a variable name like $x$, and then use keywords like 'more than' (add) or 'of' (multiply) to build the expression.