Solving Linear Equations

An equation is a balance. Whatever you do to one side must be done to the other, or the balance breaks.

Undo in reverse order

To isolate x, undo the operations in the opposite order they were applied. Addition and subtraction come off before multiplication and division.

Clear the brackets first

For 3(x + 2) = 15, either distribute to 3x + 6 = 15 or divide both sides by 3. Both are valid.

Variables on both sides

Collect the variable terms on whichever side keeps the coefficient positive. It avoids a sign error later.

Two special outcomes

If the variables vanish leaving a true statement, every number works. Leaving a false statement means no solution exists.

Always check

Substituting the answer back into the original equation catches almost every arithmetic slip.

Every step preserves the balance

Doing the same thing to both sides keeps the equation true, which is why the solution set never changes. That single principle justifies every legal move, and no other justification is needed.

Undo in reverse order

To build 3x + 5 you multiply then add; to undo it you subtract then divide. The last operation applied is the first one removed, which is the same logic as taking off shoes before socks.

Three possible outcomes

One solution, no solution when the variable cancels leaving something false, or every value when it cancels leaving something true. The variable disappearing is information, not a mistake.

Substitute into the original

Check against the equation as given, not a rearranged version — an error in the first step survives a check against the second. The substitution takes seconds and catches most slips.