How do I know which side to move the variable to?

You have a choice. Both sides are valid. Most students move the variable term with the smaller coefficient to the side with the larger one, because that keeps the coefficient of the variable positive in the end. For example, in 4x + 5 = 2x + 11, move the 2x (smaller) to the left to get 2x + 5 = 11. But you could also move the 4x to the right if you prefer — you'd get 5 = -2x + 11, solve to get -6 = -2x, then divide to get x = 3. Same answer either way.

What if I subtract the wrong term and mess up?

That's why you check. If you substitute your answer back into the original equation and it doesn't work, you know something went wrong. Go back through your steps and look for a place where you might have subtracted from only one side, or made a sign error, or forgotten to do an operation.

Do I have to move variables first and then constants, or can I do it the other way around?

You can do it either way. However, moving variables first is cleaner because after that move, you'll have a standard two-step equation. Moving constants first sometimes works too, but it can leave you with a messier-looking expression. Stick with the variable-first strategy until you're comfortable, then experiment.

What does 'collect like terms' mean exactly?

Collecting like terms means combining terms that have the same variable. Terms are 'like' if they have the exact same variable to the same power. For example, 5x and -3x are like terms, so 5x - 3x = 2x. But 5x and 5y are not like terms, and 5x and 5x^2 are not like terms either. When you move a variable term from one side of the equation to the other, you're collecting like terms — putting all the x terms together.

M8MATH-4.1

Equations with Variables on Both Sides

Learn to solve equations where the variable appears on both sides by collecting variables on one side and constants on the other.

What you'll do in this lesson

A voice-first session with the Crimsora tutor on Equations with Variables on Both Sides, then targeted practice and FRQs — with the tutor adapting to where you get stuck.

Start Equations with Variables on Both Sides free

What this lesson covers

You've already solved two-step equations like

3x + 5 = 20

. But what happens when the variable shows up on both sides, like

3x + 5 = 2x + 12

? You can't just use inverse operations anymore — you need a strategy to gather all the

x

terms together and all the numbers together. This lesson teaches you how to do that systematically, then check your answer to make sure it works.

What Does It Mean to Have Variables on Both Sides?

An equation with variables on both sides has the variable appearing in expressions on more than one part of the equation. For example, in

5x - 3 = 2x + 9

, the variable

x

appears on the left side (as

5x

) and on the right side (as

2x

). Your goal is the same as always: find the value of

x

that makes both sides equal.

The key idea is that you want to "collect like terms" — gathering all the variable terms (

x

terms) onto one side and all the constant terms (plain numbers) onto the other side. Once you do that, the equation becomes a simple two-step equation you already know how to solve.

You have a choice about which side you move things to. You can move all the

x

terms to the left and all the numbers to the right, or the other way around. Either way works. Most students find it easier to move the

x

term with the smaller coefficient, because that keeps the coefficient of

x

positive.

The Strategy: Move Variables, Then Move Constants

The process works in two main moves:

First, collect variable terms on one side. Choose which side you want the variable on. Subtract (or add) the variable term from both sides to eliminate it from one side. Remember: whatever you do to one side, you must do to the other.

Then, collect constant terms on the other side. Add or subtract the constant to move it to the other side, leaving the variable term alone.

Finally, divide to isolate the variable. You now have a two-step equation. Divide both sides by the coefficient of

x

to get

x

by itself.

Example: Solve

4x + 7 = 2x + 15

.

First, move the

2x

to the left:

4x - 2x + 7 = 2x - 2x + 15

, which gives

2x + 7 = 15

.

Next, move the

7

to the right:

2x + 7 - 7 = 15 - 7

, which gives

2x = 8

.

Finally, divide:

x = 4

.

Notice that after the first step, you had a familiar two-step equation.

Why Check Your Answer?

After you solve, always substitute your answer back into the original equation (not the simplified one) to verify both sides are actually equal. This catch mistakes you might have made while moving terms around.

Using

x = 4

4x + 7 = 2x + 15

:

Left side:

4(4) + 7 = 16 + 7 = 23

. Right side:

2(4) + 15 = 8 + 15 = 23

.

Both sides equal 23, so

x = 4

is correct.

If your answer doesn't check, go back and look for an arithmetic error or a sign mistake when you moved a term.

Common Mistakes to Avoid

One frequent error is forgetting to perform the same operation on both sides. For instance, if you write

4x + 7 = 2x + 15

and subtract

2x

from the left but forget to subtract it from the right, you'll get the wrong answer. Always keep the equation balanced.

Another mistake is mixing up signs when you move a term. If you have

4x + 7 = 2x + 15

and want to move the

+7

, you subtract 7 from both sides, turning it into a

-7

. Don't accidentally turn it into

-7

on only one side.

A third error is forgetting to divide by the coefficient at the end. Some students stop at

2x = 8

and write

x = 8

instead of

x = 4

. Always complete the division step.

Finally, many students skip checking. Checking takes 20 seconds and catches errors that cost way more time later. Make it a habit.

Key terms

Variable term.: A term containing the variable, like $3x$ or $-2a$ . The number in front is called the coefficient.
Constant term.: A plain number with no variable attached, like 5 or $-8$ .
Coefficient.: The number multiplied by the variable. In $7x$ , the coefficient is 7. In $-x$ , the coefficient is $-1$ .
Like terms.: Terms that have the same variable raised to the same power. You can combine $3x$ and $5x$ into $8x$ because they are like terms.
Collecting like terms.: Combining like terms by adding or subtracting them to simplify an expression or equation.
Inverse operation.: An operation that undoes another operation. Subtraction undoes addition; division undoes multiplication.
Substitution check.: Plugging the solution back into the original equation to verify both sides are equal.

Worked example

Solve

6x - 5 = 3x + 10

and check your answer.

Step 1: Move variable terms to one side.

I'll move the

3x

to the left side because

6x

is larger than

3x

. Subtract

3x

from both sides:

6x - 3x - 5 = 3x - 3x + 10

3x - 5 = 10

Step 2: Move constant terms to the other side.

Add 5 to both sides to move the

-5

to the right:

3x - 5 + 5 = 10 + 5

3x = 15

Step 3: Isolate the variable.

Divide both sides by 3:

x = 5

Step 4: Check by substitution.

Substitute

x = 5

into the original equation

6x - 5 = 3x + 10

:

Left side:

6(5) - 5 = 30 - 5 = 25

. Right side:

3(5) + 10 = 15 + 10 = 25

.

Both sides equal 25, so

x = 5

is correct. ✓

Practice questions

Solve

7x + 2 = 5x + 8

Answer: $x = 3$

Move the

5x

to the left by subtracting

5x

from both sides:

7x - 5x + 2 = 8

, which gives

2x + 2 = 8

. Then subtract 2 from both sides:

2x = 6

. Finally divide by 2:

x = 3

. Check:

7(3) + 2 = 21 + 2 = 23

and

5(3) + 8 = 15 + 8 = 23

. ✓

Solve

-2x + 9 = 4x - 3

Answer: $x = 2$

Move the variable term by adding

2x

to both sides (or subtracting

4x

from both sides; both strategies work). Using the first method:

-2x + 2x + 9 = 4x + 2x - 3

gives

9 = 6x - 3

. Add 3 to both sides:

12 = 6x

. Divide by 6:

x = 2

. Check:

-2(2) + 9 = -4 + 9 = 5

and

4(2) - 3 = 8 - 3 = 5

. ✓

Explain why it is important to check your answer in the original equation, not just in the simplified version. What error might you catch?

Answer: Checking in the original equation catches arithmetic errors you made while collecting terms or moving expressions from one side to the other. If you only check in a simplified version, an error from an earlier step might go unnoticed.

For example, if you correctly simplify

7x + 2 = 5x + 8

2x + 2 = 8

but then mistakenly solve to get

x = 4

(forgetting to subtract 2 first), checking in the simplified equation might pass. But checking in the original:

7(4) + 2 = 30

and

5(4) + 8 = 28

. They don't match, so you'll catch the error. This is why substituting into the original equation is a critical habit.

FAQ

How do I know which side to move the variable to?: You have a choice. Both sides are valid. Most students move the variable term with the smaller coefficient to the side with the larger one, because that keeps the coefficient of the variable positive in the end. For example, in $4x + 5 = 2x + 11$ , move the $2x$ (smaller) to the left to get $2x + 5 = 11$ . But you could also move the $4x$ to the right if you prefer — you'd get $5 = -2x + 11$ , solve to get $-6 = -2x$ , then divide to get $x = 3$ . Same answer either way.
What if I subtract the wrong term and mess up?: That's why you check. If you substitute your answer back into the original equation and it doesn't work, you know something went wrong. Go back through your steps and look for a place where you might have subtracted from only one side, or made a sign error, or forgotten to do an operation.
Do I have to move variables first and then constants, or can I do it the other way around?: You can do it either way. However, moving variables first is cleaner because after that move, you'll have a standard two-step equation. Moving constants first sometimes works too, but it can leave you with a messier-looking expression. Stick with the variable-first strategy until you're comfortable, then experiment.
What does 'collect like terms' mean exactly?: Collecting like terms means combining terms that have the same variable. Terms are 'like' if they have the exact same variable to the same power. For example, $5x$ and $-3x$ are like terms, so $5x - 3x = 2x$ . But $5x$ and $5y$ are not like terms, and $5x$ and $5x^2$ are not like terms either. When you move a variable term from one side of the equation to the other, you're collecting like terms — putting all the $x$ terms together.

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