ChemGem

How to Solve for X in an Equation (Step-by-Step, with Practice)

Solving for X (rearranging an equation) means getting the unknown alone on one side. Undo each operation with its opposite, and do it to both sides. Chemistry formulas use the same moves, so rearrange first, then put in numbers.

Spot it, solve it, practice it: 5 steps, 1 worked example, 11 practice problems. Free, no account.

ChemGemUpdated Section 0.2

The route · 5 steps

  1. Name what is being done to the unknown.
  2. Undo it with the opposite operation on both sides.
  3. Repeat until the unknown is alone.
  4. Put in the numbers with units.
  5. Check by substituting back.
Try one now See a worked example
Spot it

How to recognize solve-for-X questions

Look for "Solve" or "Solve … for" a letter: an equation where one unknown must end up alone on one side.

Example wording, with the clue words marked:

Solve: X + 8 = 15

Not sure it’s this type? Compare the neighbors:

Step-by-step

Worked example: solve-for-X

The idea

Do the same to both sides
left sideright side− 5− 5X + 512X = 7, still balanced
Whatever you do to one side, do to the other. The equation stays true.
Undo with the opposite
What is done to the unknownUndo it on both sidesExample
+ 5 is added− 5X + 5 = 12
× 3 (multiplied)÷ 33X = 18
÷ 4 (divided)× 4X/4 = 5
Unknown is in a denominator× the unknown first12/X = 3
Same rules in chemistry
  • The unknown may be X, V, n, P, m, q or M.
  • Rearrange first, then insert numbers.
  • In each example below, the highlighted operation (like − 5) is the change made to both sides.

On the periodic table: Molar mass (Mmolar, in g/mol) comes from the periodic table: add the decimal masses of the atoms in the formula (Chapter 4).

Worked example · 3 steps

Addition. Solve X + 5 = 12.

  1. 1
    Subtract 5 from both sides

    Subtracting 5 removes the added 5.

    X + 5 − 5 = 12 − 5
  2. 2
    Simplify

    X is alone.

    X = 7
  3. 3
    Check

    Put 7 back in.

    7 + 5 = 12 ✓
  4. ✓
    Answer

    X = 7

Your turn

Solve-for-X practice problems

Answer first, then check. Hints open one at a time; the answer and full solution open after you try.

Problem 1 of 11 · warm-up

Solve: X + 8 = 15

Hint 1 · the clue

Undo addition with subtraction.

Hint 2 · the next move

A number is added to X: undo addition by subtracting it from both sides.

Your path:

  1. Spot the added number
  2. Subtract it from both sides
  3. Simplify and check
Answer

Answer: X = 7

Full worked solution

What you know: X + 8 = 15. What you want: X by itself.

  1. X has 8 added to it. Undo addition with subtraction.
  2. Subtract 8 from both sides: X + 8 − 8 = 15 − 8.
  3. Simplify: X = 7.

Check: Put 7 back in: 7 + 8 = 15. It works.

Watch out: Do the same operation to both sides. Subtracting 8 only on the left breaks the equation.

Answer: X = 7

Problem 2 of 11

Solve: X − 6 = 11

Hint 1 · the clue

Undo subtraction with addition.

Hint 2 · the next move

A number is subtracted from X: undo subtraction by adding it to both sides.

Your path:

  1. Spot the subtracted number
  2. Add it to both sides
  3. Simplify and check
Answer

Answer: X = 17

Full worked solution

What you know: X − 6 = 11. What you want: X by itself.

  1. X has 6 subtracted from it. Undo subtraction with addition.
  2. Add 6 to both sides: X − 6 + 6 = 11 + 6.
  3. Simplify: X = 17.

Check: Put 17 back in: 17 − 6 = 11. It works.

Watch out: Subtracting 6 instead gives X = 5, which fails the check: 5 − 6 = −1, not 11.

Answer: X = 17

Problem 3 of 11

Solve: 5X = 35

Hint 1 · the clue

Undo multiplication with division.

Hint 2 · the next move

A number multiplies X (5X): undo multiplication by dividing both sides by that number.

Your path:

  1. 5X means 5 × X
  2. Divide both sides by 5
  3. Simplify and check
Answer

Answer: X = 7

Full worked solution

What you know: 5X = 35. What you want: X by itself.

  1. 5X means 5 × X. Undo multiplication with division.
  2. Divide both sides by 5: 5X ÷ 5 = 35 ÷ 5.
  3. Simplify: X = 7.

Check: Put 7 back in: 5 × 7 = 35. It works.

Watch out: Subtracting 5 (X = 30) is the wrong undo. The 5 is multiplying X, not added to it.

Answer: X = 7

Problem 4 of 11

Solve: X/3 = 8

Hint 1 · the clue

Undo division with multiplication.

Hint 2 · the next move

X is divided by a number (X/3): undo division by multiplying both sides by that number.

Your path:

  1. X is divided by 3
  2. Multiply both sides by 3
  3. Simplify and check
Answer

Answer: X = 24

Full worked solution

What you know: X/3 = 8. What you want: X by itself.

  1. X is divided by 3. Undo division with multiplication.
  2. Multiply both sides by 3: (X/3) × 3 = 8 × 3.
  3. Simplify: X = 24.

Check: Put 24 back in: 24 ÷ 3 = 8. It works.

Watch out: Dividing 8 by 3 is the wrong direction. To undo "÷ 3", multiply by 3.

Answer: X = 24

Problem 5 of 11

Solve: 18/X = 6

Hint 1 · the clue

First move the unknown out of the denominator.

Hint 2 · the next move

X sits in the denominator (18/X): first move it out by multiplying both sides by X, then divide.

Your path:

  1. Multiply both sides by X
  2. Now divide both sides by 6
  3. Simplify and check
Answer

Answer: X = 3

Full worked solution

What you know: 18/X = 6. What you want: X by itself.

  1. X is in the denominator. First move it out: multiply both sides by X. (18/X) × X = 6 × X, so 18 = 6X.
  2. Now 6 multiplies X. Divide both sides by 6: 18 ÷ 6 = 6X ÷ 6.
  3. Simplify: X = 3.

Check: Put 3 back in: 18 ÷ 3 = 6. It works.

Watch out: Multiplying 18 × 6 = 108 is a common slip. When X is on the bottom, the answer is 18 ÷ 6, not 18 × 6.

Answer: X = 3

Problem 6 of 11

Solve PV = nRT for P.

Hint 1 · the clue

What is multiplying P?

Hint 2 · the next move

"Solve … for P" means get P alone: V multiplies P, so divide both sides by V.

Your path:

  1. Find what multiplies P: V
  2. Divide both sides by V
  3. V cancels, leaving P alone
Answer

Answer: P = nRT/V

Full worked solution

What you know: PV = nRT. What you want: P by itself.

  1. On the left, V multiplies P. That is the only thing beside P.
  2. Undo multiplication with division: divide both sides by V. PV ÷ V = nRT ÷ V.
  3. V cancels on the left: P = nRT/V.

Check: Multiply both sides back by V: P × V = nRT. You are back to the original equation.

Why it matters: Rearrange first, then insert numbers. Chapter 9 uses this exact form.

Answer: P = nRT/V

Problem 7 of 11

Solve PV = nRT for T.

Hint 1 · the clue

Remove both factors beside T.

Hint 2 · the next move

"Solve … for T" means get T alone: n and R both multiply T, so divide by the product nR.

Your path:

  1. Find what multiplies T: n and R
  2. Divide both sides by nR
  3. nR cancels, leaving T alone
Answer

Answer: T = PV/(nR)

Full worked solution

What you know: PV = nRT. What you want: T by itself.

  1. On the right, n and R both multiply T. Both must be removed.
  2. Divide both sides by the whole product nR: PV ÷ (nR) = nRT ÷ (nR).
  3. nR cancels on the right: T = PV/(nR).

Check: Multiply both sides back by nR: nRT = PV. That is the original equation.

Watch out: Writing PV/n × R puts R on top. Keep nR together in the denominator with parentheses.

Answer: T = PV/(nR)

Problem 8 of 11

Solve M = n/V for n.

Hint 1 · the clue

Undo division by V.

Hint 2 · the next move

"Solve … for n" with n divided by V: undo division by multiplying both sides by V.

Your path:

  1. n is divided by V
  2. Multiply both sides by V
  3. V cancels, leaving n alone
Answer

Answer: n = MV

Full worked solution

What you know: M = n/V. What you want: n by itself.

  1. n is divided by V. Undo division with multiplication.
  2. Multiply both sides by V: M × V = (n/V) × V.
  3. V cancels on the right: n = MV.

Check: Divide both sides of n = MV by V: n/V = M. That is the original equation.

Watch out: Dividing M by V (n = M/V) is the wrong undo. V was dividing, so you multiply.

Answer: n = MV

Problem 9 of 11

Solve q = mcΔT for m.

Hint 1 · the clue

Which two factors multiply m?

Hint 2 · the next move

"Solve … for m" where c and ΔT multiply m: divide both sides by the whole product cΔT.

Your path:

  1. Find what multiplies m: c and ΔT
  2. Divide both sides by cΔT
  3. cΔT cancels, leaving m alone
Answer

Answer: m = q/(cΔT)

Full worked solution

What you know: q = mcΔT. What you want: m by itself.

  1. Two factors multiply m: c and ΔT. Both must be removed.
  2. Divide both sides by the whole product cΔT: q ÷ (cΔT) = mcΔT ÷ (cΔT).
  3. cΔT cancels on the right: m = q/(cΔT).

Check: Multiply back by cΔT: m × cΔT = q. That is the original equation.

Watch out: ΔT is one quantity (the change in temperature), not Δ times T. Keep c and ΔT together on the bottom.

Answer: m = q/(cΔT)

Problem 10 of 11

Solve M1V1 = M2V2 for M2.

Hint 1 · the clue

Remove the factor multiplying M2.

Hint 2 · the next move

"Solve … for M2" where V2 multiplies M2: divide both sides by V2.

Your path:

  1. Find what multiplies M2: V2
  2. Divide both sides by V2
  3. V2 cancels, leaving M2 alone
Answer

Answer: M2 = M1V1/V2

Full worked solution

What you know: M1V1 = M2V2. What you want: M2 by itself.

  1. On the right, V2 multiplies M2. Undo it with division.
  2. Divide both sides by V2: M1V1 ÷ V2 = M2V2 ÷ V2.
  3. V2 cancels on the right: M2 = M1V1/V2.

Check: Multiply back by V2: M2V2 = M1V1. That is the original equation.

Watch out: The small 1 and 2 are labels, not numbers to multiply. M1 and M2 are different quantities.

Answer: M2 = M1V1/V2

Problem 11 of 11

Solve 4X = 28.

Hint 1 · the clue

4X means 4 × X. Undo multiplication with division.

Hint 2 · the next move

A number multiplies X (4X): undo multiplication by dividing both sides by that number.

Your path:

  1. 4X means 4 × X
  2. Divide both sides by 4
  3. Simplify and check
Answer

Divide both sides by 4: X = 7.

Full worked solution

What you know: 4X = 28. What you want: X by itself.

  1. 4X means 4 × X. Undo multiplication with division.
  2. Divide both sides by 4: 4X ÷ 4 = 28 ÷ 4.
  3. Simplify: X = 7.

Check: Put 7 back in: 4 × 7 = 28. It works.

Watch out: Subtracting 4 (X = 24) is the wrong undo. The 4 multiplies X; it is not added.

Answer: Divide both sides by 4: X = 7.

Watch out

Common solve-for-X mistakes