Equations with Variables on Both Sides (cont)
Once you can solve , the next level is the same equation wearing a disguise: . Parentheses on one or both sides don't change the strategy — they just add one job to the front of it. Distribute first, then it's an equation you already know how to solve.
The full order of operations for these problems: distribute, combine like terms on each side, gather the variables onto one side, then finish with the usual two steps. Every equation in this lesson follows that exact script.
Distribute first, then gather
If an equation has parentheses, distribute before anything else. In , multiplying through gives . Now the parentheses are gone and it's a standard variables-on-both-sides problem.
To gather the variables, subtract one of the variable terms from both sides. Subtracting the smaller one — here — keeps the remaining coefficient positive: . Then subtract and divide by to get .
Distributing a negative
The equation hides the most common error in this topic. Distributing flips both signs inside: , not . A negative outside the parentheses changes the sign of every term inside.
After distributing, the solve is routine: . Add to both sides to get , add to get , and divide by : .
Parentheses on both sides
Some equations distribute on both sides, like . Handle each side independently — left becomes , right becomes — and then gather as usual.
Whatever the equation looks like, the finish line is the same: variable terms on one side, plain numbers on the other, one division at the end.
Worked examples
Example 1: distribute on one side
Solve .
Answer:
Example 2: distribute a negative
Solve .
Answer:
Example 3: distribute on both sides
Solve .
Answer:
Try one yourself
Common questions
Which side should I move the variable to?
Either side works, but moving the smaller variable term keeps the coefficient positive, which means fewer sign mistakes. In , subtract rather than .
Do I have to distribute first? Can I divide both sides instead?
Sometimes dividing works — in you could divide both sides by . But if the other side doesn't divide cleanly, or there are variable terms on it, distributing is the move that always works. When in doubt, distribute.
What if I get the variable on the right, like ?
That's fine. and say the same thing. Divide by and you have either way.
How do I check my answer?
Substitute it into the original equation — parentheses and all — and simplify each side separately. If both sides land on the same number, the solution is correct.
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