1. The Balance Scale Method (Inverse Operations):
Whatever operation is attached to the variable, undo it with the opposite operation on both sides:
• Undo addition with subtraction ($+ 5 o - 5$)
• Undo multiplication with division ($ imes 3 o \div 3$)
Example: $4x + 7 = 31 \implies 4x = 24 \implies x = 6$.

2. Reversing Function Machines (Working Backwards):
When given the final output of a multi-stage machine, write out the operations in reverse order, inverting each arithmetic sign as you go.

3. Symbol Elimination & Substitution:
When two equations feature icons or objects (e.g. 2 Apples + 1 Banana = 22; 1 Apple + 1 Banana = 15), subtract Equation 2 from Equation 1 to find the value of 1 Apple instantly ($22 - 15 = 7$). Then substitute back!

4. The nth Term Sequence Rule:
For any linear sequence (e.g. $4, 7, 10, 13\dots$):
1. Find the step difference: $+3 \implies 3n$.
2. Subtract difference from 1st term to find the zero-term adjustment: $4 - 3 = +1$.
3. Formula: $3n + 1$.

⚠️ Trap 1: The Minus-Before-x Sign Error:
In $15 - 2x = 7$, students often write $2x = 15 + 7 = 22$.
Fix: Add $2x$ to both sides first: $15 = 7 + 2x \implies 8 = 2x \implies x = 4$.

⚠️ Trap 2: Reversing Order of Operations:
When solving $3x + 5 = 26$, do not divide by 3 first! Always peel off additions/subtractions before division/multiplication.

⚠️ Trap 3: The Multiplied Letter Confusion:
In $4a$, if $a = 5$, the answer is $4 imes 5 = 20$, never 45!

⚠️ Trap 4: Bracket Distributive Oversight:
In $3(x + 4) = 27$, remember that 3 multiplies both terms inside ($3x + 12 = 27$), OR divide both sides by 3 immediately ($x + 4 = 9 \implies x = 5$).