📋 Prerequisites

• Order of Operations & Arithmetic
• Fractions & Decimals

Variables are what turn arithmetic into algebra. A variable is just a letter that stands for a number we don't know yet. Learning to translate words into equations is the single most important skill for competition math — nearly every problem eventually becomes "solve for $x$."

📚 Key Concepts

What Is a Variable?

• A variable is a letter (usually $x$, $y$, $n$) that represents an unknown number
• An expression is a mathematical phrase with variables and numbers: $ 3x + 5 $
• An equation says two expressions are equal: $ 3x + 5 = 17 $
• To evaluate an expression, plug in the value of the variable

Example: If $ x = 4 $, then $ 3x + 5 = 3(4) + 5 = 12 + 5 = 17 $

Solving One-Step Equations

To solve for a variable, isolate it by doing the same thing to both sides:

• If $ x + a = b $, subtract $a$: $ x = b - a $
• If $ x - a = b $, add $a$: $ x = b + a $
• If $ ax = b $, divide by $a$: $ x = \frac{b}{a} $
• If $ \frac{x}{a} = b $, multiply by $a$: $ x = ab $

The golden rule: whatever you do to one side, you must do to the other side.

Solving Multi-Step Equations

For harder equations, follow this order:

1. Distribute if there are parentheses: $ 2(x + 3) \to 2x + 6 $
2. Combine like terms on each side separately
3. Get all $x$ terms on one side, all numbers on the other
4. Divide to get $x$ alone
5. Check your answer by plugging back in!

Translating Words to Equations

This is the most important skill. Here's the translation dictionary:

• "is", "equals", "gives" \(\\to\) $ = $
• "sum of", "more than", "increased by" \(\\to\) $ + $
• "difference", "less than", "decreased by" \(\\to\) $ - $
• "product", "times", "of" \((with fractions) \\to\) $ \times $
• "quotient", "divided by" \(\\to\) $ \div $
• "twice", "double" \(\\to\) $ 2 \times $
• "half of" \(\\to\) $ \frac{1}{2} \times $

Trick: "5 less than $x$" = $ x - 5 $ (not $ 5 - x $). The "than" reverses the order.

✏️ Example Problems