Variables on Both Sides

Variables on Both Sides

When a variable appears on both sides of an equation, your first job is to get all the variable terms onto one side and all the constant terms onto the other. Do this by adding or subtracting a variable term from both sides, the same way you would move a constant.

Strategy

1. Simplify each side first (distribute, combine like terms) if needed.
2. Choose one side to eliminate the variable from — typically the side with the smaller coefficient — and add/subtract that variable term from both sides.
3. Solve the resulting one-variable equation using the usual two-step method.

Example

Solve 5x - 8 = 2x + 13.

1. Subtract 2x from both sides to collect the variable terms on the left: 5x - 2x - 8 = 13 \;\Rightarrow\; 3x - 8 = 13
2. Add 8 to both sides: 3x = 21
3. Divide both sides by 3: x = 7

Check: 5(7) - 8 = 35 - 8 = 27 and 2(7) + 13 = 14 + 13 = 27. Both sides agree, so x = 7 is correct.

Special Case: No Solution

Sometimes, after collecting variable terms, they cancel out entirely and leave a false statement. For example:

Subtracting 2x from both sides gives 5 = -3, which is never true. This means the equation has no solution — there is no value of x that works.

Special Case: Infinitely Many Solutions

Other times, the variable terms cancel and leave a true statement, such as:

Distributing gives 3x + 6 = 3x + 6, and subtracting 3x from both sides gives 6 = 6, which is always true. This means every value of x is a solution — the two sides are actually the same expression written differently.

Recognizing These Cases Early

You don't need to wait until the very end to notice a no-solution or infinite-solutions case. After distributing and combining like terms on each side, compare the coefficients of the variable term. If the coefficients are equal on both sides, the variable will vanish once you subtract — at that point, whether you get a true or false statement about the remaining constants tells you immediately which special case you're in. This lets you predict the outcome before doing all the remaining arithmetic.

Key Takeaways

• Collect variable terms on one side by adding/subtracting a variable term from both sides.
• After collecting variables, finish with the standard two-step method.
• If the variable disappears and leaves a false statement (like 5=-3), there is no solution.
• If the variable disappears and leaves a true statement (like 6=6), every real number is a solution.