How to Solve Equations With Variables on Both Sides (Step by Step)

To solve an equation with variables on both sides, move all variable terms to one side and all constant terms to the other, then divide by the coefficient of the variable. For example, 5x + 3 = 2x + 12 becomes 3x = 9, so x = 3.

The trick is choosing which side gets the variables. Most students pick the side with the larger coefficient so they end up with a positive number in front of x. Below we walk through two full examples and the exact steps to follow.

A fully worked example

Solve: 5x + 3 = 2x + 12

1. Subtract 2x from both sides to collect the variables on the left: 5x - 2x + 3 = 12, which simplifies to 3x + 3 = 12.
2. Subtract 3 from both sides to isolate the variable term: 3x = 9.
3. Divide both sides by 3: x = 3.
4. Check by substituting into the original: 5(3) + 3 = 18 and 2(3) + 12 = 18. Both sides match, so x = 3 is correct.

The step-by-step method

1. Simplify each side first. Distribute any parentheses and combine like terms. Example: 3(x + 2) + x = 4x + 6 becomes 4x + 6 = 4x + 6.
2. Move all variable terms to one side. Add or subtract to eliminate the variable from the side with the smaller coefficient. Example: from 7x = 4x + 9, subtract 4x to get 3x = 9.
3. Move all constants to the other side. Add or subtract so only the variable term remains on one side. Example: 3x + 5 = 14 becomes 3x = 9.
4. Divide by the coefficient of the variable. This isolates x. Example: 3x = 9 gives x = 3.
5. Check your answer. Substitute back into the original equation to confirm both sides are equal.

Special cases to watch for

• If the variables cancel and you're left with a true statement like 6 = 6, the equation has infinitely many solutions.
• If the variables cancel and you're left with a false statement like 3 = 7, the equation has no solution.

A second worked example with distribution

Solve: 4(x - 2) = 2x + 6

1. Distribute the 4 on the left: 4x - 8 = 2x + 6.
2. Subtract 2x from both sides: 2x - 8 = 6.
3. Add 8 to both sides: 2x = 14.
4. Divide both sides by 2: x = 7.
5. Check: 4(7 - 2) = 4(5) = 20 and 2(7) + 6 = 20. Correct.

Common mistakes to avoid

• Forgetting to distribute to every term. In 4(x - 2), you must multiply the 4 by both the x and the -2. Writing 4x - 2 is a common slip.
• Sign errors when moving terms. When you move 2x across the equals sign, it becomes -2x. Do the same operation on both sides to stay safe.
• Combining unlike terms. You cannot add 3x and 5 into 8x. Variables and constants stay separate.
• Dividing only one side by the coefficient. Whatever you do to one side, you must do to the other. Dividing 3x = 9 by 3 gives x = 3, not x = 9.
• Skipping the check. A quick substitution catches most arithmetic mistakes before they cost points.

Now that you have seen the method, the fastest way to lock it in is to work a few problems yourself with feedback at each step. Sign up for LernOS to try our guided tutor — it walks you through problems like these one question at a time, so you catch mistakes as you make them rather than at the end.