Solving linear equations: keep the balance, not a bag of tricks

Start by reducing clutter

Distribute and combine like terms on each side before moving terms across the equals sign. This exposes the equation's real structure.

Fractions can often be cleared by multiplying every term on both sides by a common denominator. Skipping a term breaks the balance.

Isolate in a sensible order

Move variable terms to one side and constants to the other. Then undo multiplication or division around the variable.

There is rarely one mandatory sequence. Prefer moves that keep coefficients small and reduce the chance of sign errors.

The check is part of the solution

Substitute the proposed value into the original equation, not a later simplified line. Both sides should produce the same number.

A check catches arithmetic slips and also reveals equations with no solution or infinitely many solutions when the variable disappears.

Worked example

Common mistakes

• Moving a term by changing its sign without naming the operation.
• Applying a denominator-clearing multiplier to only some terms.
• Checking in a simplified line that already contains the same error.

Keep these ideas

• Simplify each side first.
• Use the same operation on both sides.
• Check in the original equation.