Linear equations stay balanced

The idea

An equation is a promise that two quantities are equal. Whatever operation you apply to one side, apply to the other; solving means undoing operations until the unknown stands alone.

Real-world: splitting a restaurant bill

Four friends share one fixed table fee of $12 plus four identical meals, and the total is $92. With meal price $x$: $4x + 12 = 92$. Removing the fee from both sides gives $4x = 80$, so each meal costs $20.

Worked example

Solve by applying the same inverse operation to both sides:

Check by substitution: $3(4)+6=18$ ✓

Cheat sheet

$3x + 6 = 18 \;\to\; 3x = 12 \;\to\; x = 4$

• Same operation, both sides, always.
• Undo in reverse order: additions before multiplications.
• Always verify by substituting back.