Writing and Interpreting Equations
Before you can solve an equation, you have to be able to write one. Word problems hand you a sentence — "four times a number, decreased by 9, is 27" — and your job is to translate it into symbols: . Going the other direction matters just as much: given an equation like , you should be able to say what the and the mean in the story.
The translation is mostly vocabulary. A handful of words map to the same symbols every time, and once you know the map, writing equations stops feeling like guesswork.
The word-to-symbol map
Sum, increased by, and more than all mean addition. Difference, decreased by, and less than mean subtraction. Times, of, and product mean multiplication. Quotient and per mean division. And the single most important word: is means equals. When you see "is," that's where the sign goes.
"A number" means you pick a variable. If the problem doesn't name one, use or — it doesn't matter which letter, as long as you use it consistently.
Translate left to right, phrase by phrase. "Four times a number" becomes . "Decreased by 9" takes away: . "Is 27" finishes it: .
The "less than" trap
"Six less than twice a number" is not . "Less than" flips the order: the is being taken away from the other quantity, so the correct translation is . The same goes for "less than" cousins like "fewer than" and "subtracted from."
A quick check: "six less than ten" is , and you compute it as — the number after "less than" comes second. Keep that tiny example in your head and you'll never flip it.
Interpreting an equation in context
Real-world equations usually have two kinds of numbers: a rate that multiplies the variable, and a one-time amount that stands alone. In , where is the total cost of a plumber visit lasting hours, the is attached to — every extra hour adds dollars, so it's the cost per hour. The never changes no matter how long the visit lasts, so it's the one-time fee.
To interpret any number in an equation, ask: does this amount repeat with the variable, or does it happen once? The multiplier repeats; the lone constant happens once.
Worked examples
Example 1: a phrase with two operations
Write an equation for "Three times a number, increased by 7, is 25." Use for the number.
Answer:
Example 2: a "less than" phrase
Write an equation for "Five less than half a number is 9." Use for the number.
Answer:
Example 3: interpreting a number in context
A phone plan's monthly cost is , where is the gigabytes of data used. What does the represent?
Answer: The is the cost per gigabyte of data.
Try one yourself
Common questions
Does it matter which letter I use for the variable?
No. , , and all work the same. Some problems tell you which letter to use — then use that one. Otherwise pick a letter, say what it stands for, and stick with it.
How do I know where the equals sign goes?
Look for the word "is" (or "was," "will be," "equals," "gives," "totals"). Everything before it goes on one side, everything after it goes on the other.
Why is "six less than twice a number" and not ?
"Less than" reverses the order you hear the numbers in. Six less than something means you start with that something and take away from it, so the is written second: .
Do I have to solve the equation too?
Only if the problem asks. Many questions only ask you to write the equation or to explain what a number in it means — read the question stem carefully before doing extra work.
Want the video version?
Allday Everyday Math has video lessons, practice, and an AI tutor for every topic, Pre-Algebra through Algebra 2.