Variables on Both Sides

When x appears on both sides, gather them together first.

7x − 4 = 3x + 12. Subtract 3x from both sides: 4x − 4 = 12. Then 4x = 16, so x = 4.

Tip: move the variables to whichever side has the bigger coefficient. That keeps the number in front positive and avoids one sign error.

Two odd endings can happen. If you end at something true like 5 = 5, every number works — infinitely many solutions. If you end at something false like 3 = 8, nothing works — no solution. Neither is a mistake; both are real answers.

Collecting like terms across the equality uses the addition property of equality, and choosing the side that leaves a positive leading coefficient is purely error-reduction — mathematically either choice is fine.

The degenerate outcomes classify the equation. An identity (2(x+3) = 2x+6) reduces to 0 = 0, true for all reals, so the solution set is ℝ. A contradiction (2x + 1 = 2x + 5) reduces to 0 = 4, so the solution set is ∅. Geometrically these are two lines that coincide and two parallel lines respectively — the same trichotomy that reappears with systems of equations, which is not a coincidence: solving f(x) = g(x) is finding where two lines intersect.

Reporting these correctly matters. "No solution" and "the solution is 0" are entirely different claims, and writing 0 when the equation is a contradiction is a substantive error rather than a notational one.