The Golden Rule

Whatever you do to one side of an equation, do the same to the other side. This maintains equality. To solve 3x + 5 = 14: subtract 5 from both sides → 3x = 9. Divide both sides by 3 → x = 3. Always isolate x by working backwards through the order of operations: undo addition/subtraction first, then undo multiplication/division.

Equations with Variables on Both Sides

For 5x + 3 = 2x + 12: first collect all x terms on one side. Subtract 2x from both sides: 3x + 3 = 12. Then subtract 3: 3x = 9. Divide by 3: x = 3. The strategy: move smaller x coefficient to the other side to keep x coefficient positive, making the arithmetic easier to follow.

Equations with Fractions

Multiply every term by the LCM of all denominators to eliminate fractions first. For x/3 + 2 = 5: multiply everything by 3 → x + 6 = 15 → x = 9. For x/2 + x/3 = 5: LCM is 6, multiply by 6 → 3x + 2x = 30 → 5x = 30 → x = 6. Eliminating fractions at the start avoids errors with fraction arithmetic during solving.

Always Verify Your Answer

After solving, substitute your answer back into the original equation to check it satisfies both sides. For 3x + 5 = 14 with x = 3: left side = 3(3) + 5 = 9 + 5 = 14 = right side ✓. Verification catches arithmetic errors and builds confidence. In exams, showing verification often earns an additional mark even if the answer is wrong.

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