Linear Equations - Rules and Formulas

Linear Equations - Rules and Formulas

Quick Reference Card

The Balance Principle

> Whatever you do to one side of the equation, you must do to the other.

Basic Transformations

Operation	Equation	Becomes
Add $a$	$x + a = b$	$x = b - a$
Subtract $a$	$x - a = b$	$x = b + a$
Multiply by $a$	$x / a = b$	$x = ab$
Divide by $a$	$a x = b$	$x = b / a$

Moving Terms (Change Side, Change Sign)

When a term crosses the equals sign, it changes sign:

x + 5 = 12 \Rightarrow x = 12 - 5

x - 3 = 7 \Rightarrow x = 7 + 3

2 x = 10 \Rightarrow x = 10/2

Parentheses

Use distributive property:

a (b + c) = ab + a c

Example: $3 (x + 4) = 3 x + 12$

Fractions

Unknown in numerator

\frac{x}{a} = b \Rightarrow x = ab

Unknown in denominator

\frac{a}{x} = b \Rightarrow x = \frac{a}{b}

General fraction

\frac{a x}{b} = c \Rightarrow a x = b c \Rightarrow x = \frac{b c}{a}

Number of Solutions

Situation	Result
$a x = b$ , $a \neq = 0$	One solution: $x = b / a$
$0 x = b$ , $b \neq = 0$	No solution
$0 x = 0$	Infinite solutions

Solving Checklist

✓ Expand any parentheses
✓ Collect like terms on each side
✓ Move all $x$ terms to one side
✓ Move all number terms to the other side
✓ Simplify both sides
✓ Divide/multiply to isolate $x$
✓ Check your answer

Common Mistakes

Wrong	Correct
$x + 5 = 12 \Rightarrow x = 12$	$x = 12 - 5 = 7$
$x - 3 = 7 \Rightarrow x = 7 - 3 = 4$	$x = 7 + 3 = 10$
$\frac{x}{3} = 6 \Rightarrow x = 6 \div 3 = 2$	$x = 6 \times 3 = 18$

Comprehensive Guide - Full explanations
Simple Equations - Practice examples