Multi-Step Equations

Variables on Both Sides

a₁x + b₁ = a₂x + b₂

When the unknown x appears on both sides of the equation, the goal is to gather all x-terms on one side and all constants on the other — then divide.

Step 1: Subtract the smaller x-coefficient from both sides so the variable ends up on the left.

Step 2: Move the constant to the right by adding or subtracting it from both sides.

Step 3: Divide both sides by the remaining coefficient of x.

Step 4: Substitute your answer back into the original equation to verify both sides match.

Always perform the same operation on both sides — the equation is a balance scale.

Special Cases

a₁ − a₂ = 0 → check constants

After moving x-terms to one side, if the coefficient of x becomes zero, two situations arise:

No Solution — the constants are not equal (e.g., 0 = 5). The lines are parallel and never intersect. Write: no solution or ∅.
All Real Numbers — the constants are also equal (e.g., 0 = 0). Every value of x satisfies the equation. The two sides are the same line. Write: x ∈ ℝ.

Think of it graphically: two lines either intersect once, are parallel, or are identical.