Video Lecture 1: The Balance of Algebra — Solving Linear Equations with Pan Scales

# Video Lecture 1: The Balance of Algebra — Solving Linear Equations with Pan Scales
**Speaker:** Prof. Arvind Swaminathan | **Duration:** ~05:50 mins

### Introduction: What is an Equation?
An equation is a statement of equality containing one or more unknown variables:
$$2x - 3 = 7$$
Think of an equation as a **two-pan balance scale**:
- Left Pan $= 2x - 3$
- Right Pan $= 7$
The "=" sign asserts that the scale is in **perfect horizontal balance**.

### The Golden Rule of Balance:
To keep the scale balanced, whatever mathematical operation you perform on the left pan, you **MUST perform the exact same operation on the right pan**:
1. Add $3$ to both sides: $2x - 3 + 3 = 7 + 3 \implies 2x = 10$.
2. Divide both sides by $2$: $\frac{2x}{2} = \frac{10}{2} \implies \mathbf{x = 5}$.

### The Transposition Method (High-Speed Algebra):
Instead of writing the balance steps every time, we transpose terms across the equality sign:
- When a term moves to the other side:
- Addition ($+$) becomes Subtraction ($-$).
- Subtraction ($-$) becomes Addition ($+$).
- Multiplication ($\times$) becomes Division ($\div$).
- Division ($\div$) becomes Multiplication ($\times$).