Foundations — Word problems using linear equations

Before you translate a single sentence, you need to know what the pieces mean. Below is every symbol and idea the parent topic Word Problems Using Linear Equations leans on — built from nothing, each one resting on the one before it.

1. A number, and a number line

Everything starts with a number: how many apples, how much money, how many years.

The picture for a number is a single dot on a number line — a straight road with $0$ in the middle, positive numbers marching right, negative numbers marching left. "How old are you?" is a dot somewhere to the right of $0$.

Why the topic needs it: age, money, distance — every answer we ever find lives as a single dot on this line. The left side of the line (the negative region) will later tell us an answer is nonsense (you cannot be $-5$ years old).

2. The unknown letter $x$

The hidden number in a story doesn't yet have a value — so we cannot draw its dot. Instead we draw a box with a question mark and label it $x$.

Read more about this in Linear Equations in One Variable.

3. The four operations, in words and pictures

Stories describe changes to quantities. Each change is one of four operations. Here is the plain-word meaning, the picture, and the English words that trigger it.

Why the topic needs it: every sentence in a word problem is secretly one of these four actions. "Father is 3 times as old" is $\times 3$. "In 12 years" is $+12$. Full details in Inverse Operations and Algebraic Expressions.

4. Writing a number next to a letter means multiply

Before we build bigger expressions, one silent convention must be spoken aloud.

5. Building an expression from $x$

Now we can describe other unknown quantities in the same story. "The father is 3 times as old as the son" becomes $3x$. "The son in 12 years" becomes $x+12$. A bundle of a letter and numbers joined by operations is an algebraic expression.

6. The equals sign $=$ and the idea of balance

Now the key symbol. $=$ does not mean "the answer is". It means "these two sides weigh exactly the same" — like a see-saw that is perfectly level.

The trigger words for $=$ in a story: "is", "equals", "will be", "same as", "in total". When you spot one of these, that sentence is your equation.

7. Keeping the balance: inverse operations

To find $x$, we peel away everything wrapped around it — but we must keep the scale level. So whatever we do to the left pan, we do to the right pan. To undo $+$, we use $-$. To undo $\times$, we use $÷$. These undo-pairs are inverse operations.

8. Ratios and rates (for the harder types)

Two special ideas power the money, mixture, distance, and work problems.

First, a word on the rupee symbol: ₹ is just India's mark for money, the way "$" marks dollars. "₹5" means five rupees. Nothing mathematical — it only labels a number as money.

How these foundations feed the topic

Read the map below from top to bottom: each box is one idea from this page, and the arrows show what it unlocks. Numbers and the operations combine into expressions; expressions plus the equals-sign become a linear equation; inverse operations let us solve it — and ratios/rates supply the numbers for the trickier stories. Everything funnels into the bottom box, the parent topic itself.

The topic Applications of Algebra and its bigger sibling Simultaneous Linear Equations reuse every one of these pieces — so mastering them here pays off twice.

Equipment checklist

Say each answer out loud before revealing it. If any is fuzzy, reread that section.