Writing Expressions with Variables

An algebraic expression uses numbers, operation symbols, and at least one variable to describe a quantity without saying exactly what the number is. Writing expressions with variables means turning a sentence, such as "five more than a number," into math, such as \(n + 5\). This skill builds directly on knowing what a variable is, since every expression starts by choosing a letter to stand for the unknown quantity.

A variable is a placeholder letter, like \(x\), \(n\), or \(c\), that stands for a number we don't know yet or a number that can change. Instead of writing a new equation for every possible value, we write one algebraic expression that works no matter what number the variable represents. This is the foundation you'll need later when you start solving equations, since solving equations with variables begins by writing the correct expression first.

The fastest way to translate a word phrase is to look for a keyword. Certain words almost always mean addition, subtraction, multiplication, or division. The chart below lists the most common ones.

Follow the same three steps for any word phrase:

1. Find the unknown quantity and choose a letter to represent it.
2. Look for a keyword to identify the operation (addition, subtraction, multiplication, or division).
3. Write the numbers, the variable, and the operation symbol in the correct order.

Example 1: A simple algebraic phrase

Write an algebraic expression for "a number increased by 8."

"Increased by" signals addition. Let the unknown number be \(x\). The expression is \(x + 8\).

Example 2: The quotient of 9 and c

Write an expression for the quotient of 9 and \(c\).

"Quotient" means division, and it must be written in the order the words give it: 9 divided by \(c\). The expression is \(9 \div c\), which can also be written as \(\frac{9}{c}\).

Example 3: Two expressions with the same solution

Sometimes a problem asks for two different expressions that both give the same value. For example, write two expressions where the solution is 41.

You could write \(35 + 6\) or \(50 - 9\). Both expressions use different operations and numbers, but they both equal 41. This shows that many different expressions can represent the same quantity.

Example 4: The perimeter of a figure

A rectangle has a length of \(l\) and a width of \(w\). Write two expressions for its perimeter.

One way is to add all four sides: \(l + l + w + w\). A shorter way is to use multiplication: \(2l + 2w\). Both expressions describe the exact same perimeter, just written differently.

Once an expression is written, test it by substituting a number for the variable. If the phrase was "5 less than a number" and you write \(n - 5\), try \(n = 10\): \(10 - 5 = 5\), which matches "5 less than 10." If the numbers don't make sense with the words, re-check the order of operations, since subtraction and division depend on the order the words appear.

The most frequent error is reversing subtraction or division. "5 less than a number" is \(n - 5\), not \(5 - n\). Similarly, "the quotient of 9 and c" is \(9 \div c\), not \(c \div 9\). Always write the phrase in the same order it's spoken, and double-check with a substituted value if you're unsure.

Related lessons

• Ca Grade 5