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How to Solve Equations With Variables on Both Sides

An equation with x on both sides is not a new kind of problem. It is an ordinary equation with one extra move at the start: get every x term onto one side first, and you are back to something you already know how to finish.

The method

  • Clear any brackets first, distributing carefully including the sign.
  • Move the variable terms to one side by adding or subtracting the same term on both sides.
  • Move the constants to the other side the same way.
  • Divide by the coefficient of the variable.
  • Substitute back into the original equation to check.

Which side should the variables go to?

Whichever side keeps the coefficient positive. In 4x = 15x - 4, moving 4x to the right gives 11x rather than -11x, and one less sign to lose track of. It is not required, it just makes the arithmetic quieter.

Why the check matters here

This topic is where sign errors are born, and a sign error produces an answer that looks perfectly reasonable. Substituting back is the only step that catches it, and it takes ten seconds.

Worked examples

Example 1

Question

Solve: 5x + 7 = 2x + 19

Answer

x = 4

Steps

  1. 1Subtract 2x from both sides to gather the variables: 3x + 7 = 19.
  2. 2Subtract 7 from both sides: 3x = 12.
  3. 3Divide both sides by 3: x = 4.
  4. 4Check: 5(4) + 7 = 27, and 2(4) + 19 = 27. Both sides agree.

Example 2

Question

Solve: 4x = 15x - 4

Answer

x = 4/11

Steps

  1. 1Move the 4x to the right to keep the coefficient positive: 0 = 11x - 4.
  2. 2Add 4 to both sides: 4 = 11x.
  3. 3Divide by 11: x = 4/11.
  4. 4Check: 4(4/11) = 16/11, and 15(4/11) - 4 = 60/11 - 44/11 = 16/11. Both sides agree.

A fractional answer is not a sign you did something wrong. Leave it as a fraction unless the question asks for a decimal.

Example 3

Question

Solve: 8 - 3x = 4x + 22

Answer

x = -2

Steps

  1. 1Add 3x to both sides to move the variable term: 8 = 7x + 22.
  2. 2Subtract 22 from both sides: -14 = 7x.
  3. 3Divide by 7: x = -2.
  4. 4Check: 8 - 3(-2) = 8 + 6 = 14, and 4(-2) + 22 = -8 + 22 = 14. Both sides agree.

Subtracting a negative is where this one bites. 8 - 3(-2) is 8 + 6, not 8 - 6.

Common mistakes

Moving a term across without changing its sign: 5x + 7 = 2x + 19 becoming 3x + 7 = 19 + 2x.

Whatever you do to one side you do to the other. Subtracting 2x removes it from the left and from the right.

Distributing only into the first term: 3(x - 4) becoming 3x - 4.

The multiplier hits every term inside the bracket: 3x - 12.

Dividing only the variable term at the end, leaving 3x = 12 as x = 12.

Divide both sides by 3. The right side becomes 4.

Stopping at -14 = 7x and reading the answer as -14.

Finish the division. x = -2, and the check confirms it.

Practice it

Work these out first, then open the answer. Stuck halfway? Select the problem and ask Solvecoon for the step you are missing.

  • 1. Solve: 6x - 5 = 2x + 11

    Show answer

    x = 4. Subtract 2x to get 4x - 5 = 11, then add 5 and divide.

  • 2. Solve: 9 - 2x = 5x - 12

    Show answer

    x = 3. Add 2x to both sides to get 9 = 7x - 12, then add 12.

  • 3. Solve: 3(x + 2) = x - 4

    Show answer

    x = -5. Distribute first: 3x + 6 = x - 4, so 2x = -10.

Questions people ask

Can I move the variables to the right instead of the left?
Yes. The answer is identical. Choose the side that leaves you with a positive coefficient, because it means fewer signs to track.
What if the variables cancel out completely?
If you are left with something true like 5 = 5, every value of x works. If you are left with something false like 5 = 7, there is no solution. Both are legitimate answers.
Do I have to check my answer?
The exam will not always give a mark for it, but it catches the sign error that would cost you the whole question. Ten seconds is a good trade.
What if there are fractions in the coefficients?
Multiply every term on both sides by the common denominator first. That clears the fractions and leaves an ordinary equation.
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