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Level 10Grade 10skill: two_step_algebra· 8 min read

Two-Step Algebra Equations

In short: A two-step equation hides the variable behind two operations. Undo them in reverse order: clear the added or subtracted constant first, then divide out the coefficient — doing the same to both sides each time. The classic trap is dividing before subtracting, or dropping a negative sign; get the order and signs right and these become routine.

How to solve a two-step equation

A two-step equation hides your variable behind exactly two operations. In 3x + 7 = 22, the x was first multiplied by 3, then had 7 added. Your job is to strip those two operations away until x stands alone — subtract 7 to reach 3x = 15, then divide by 3 to find x = 5.

Here is the whole trick, and it is the thing most students get backwards: you undo the operations in the reverse of the order they were applied. Multiplication happened, then addition. So you undo the addition first, then the multiplication. That is BODMAS running backwards.

Every move you make happens to both sides of the equals sign. That is what keeps the equation balanced, and it is the only rule you are ever really using.

  1. Look at the variable and name the two operations attached to it, and the order they were applied.
  2. Undo the operation that was applied LAST (usually the added or subtracted constant) by doing its inverse to both sides.
  3. Undo the operation that was applied FIRST (usually the coefficient) by doing its inverse to both sides.
  4. Watch the sign: a term crossing the equals sign flips, and dividing by a negative flips the sign of the answer.
  5. Substitute your answer back into the original equation to confirm both sides match.

Why reverse order is the whole game Why it works

Built vs. unbuilt
The order of operations tells you the order an expression was BUILT. To take it apart, you run that order in reverse.

3x + 7 was built as: take x, multiply by 3, then add 7. Unbuild it: subtract 7 first, then divide by 3.

Outer layer first
Undo the operation furthest from the variable before the one closest to it.

In (x + 5)/3 = 8, the last thing done was dividing by 3, so multiply by 3 first: x + 5 = 24, then subtract.

Sign discipline
When the variable term is negative, isolate it, then divide by the negative coefficient — the answer's sign flips.

9 − 4x = 33 becomes −4x = 24, and dividing by −4 gives x = −6, not +6.

Think of it as unwrapping a package: whatever was taped on last comes off first. Get the order right and two-step equations stop being guesswork — you are just reversing a recipe you can already read.

Worked examples

Solve 7x − 4 = 45. = x = 7
  1. The last operation was subtracting 4, so undo it first: add 4 to both sides.
  2. 7x = 49.
  3. Now undo the multiplication: divide both sides by 7.
  4. x = 7.
Solve 9 − 4x = 33. (The variable term is negative — watch the sign.) = x = −6
  1. Subtract 9 from both sides to isolate the variable term: −4x = 24.
  2. The coefficient is −4, so divide both sides by −4.
  3. A positive divided by a negative is negative: x = −6.
  4. Check: 9 − 4(−6) = 9 + 24 = 33. Correct.
Solve (x + 5)/3 = 8. (Structure matters here.) = x = 19
  1. The last operation applied was dividing by 3, so undo it first: multiply both sides by 3.
  2. x + 5 = 24.
  3. Now subtract 5 from both sides.
  4. x = 19. (Subtracting 5 first, before multiplying, would wrongly give 9.)
A streaming plan charges a one-time $12 signup fee plus a fixed monthly rate. After 5 months, Mia has paid $47 in total. Find the monthly rate. = $7 per month
  1. Let m be the monthly rate. Total cost: 5m + 12 = 47.
  2. Undo the added fee first: subtract 12 from both sides, giving 5m = 35.
  3. Undo the multiplication: divide both sides by 5.
  4. m = 7, so the monthly rate is $7.
Solve 40 = 3x − 8. (The variable is on the right — don't rush.) = x = 16
  1. The last operation on the right was subtracting 8, so add 8 to both sides: 48 = 3x.
  2. Divide both sides by 3.
  3. x = 16. (Dividing 40 by 3 first is the tempting mistake — but −8 must be cleared first.)
  4. Check: 3(16) − 8 = 48 − 8 = 40. Correct.

Common mistakes

Stopping after the first step

✗ For 4x − 9 = 19, adding 9 gives 4x = 28 — and the student writes 28 as the answer.

You have only isolated the variable term, not the variable. 4x = 28 is a halfway house; x is still multiplied by 4.

Fix: Finish the job: divide both sides by 4 to get x = 7. Always ask, 'Is my variable completely alone?' before writing an answer.

partial_computation

Dividing before subtracting

✗ For 3x + 12 = 30, dividing 30 by 3 first gives 10, then subtracting 12 gives −2.

You undid the operations in the wrong order. The 12 was added last, so it must be removed first. Dividing first only works if you divide the entire equation, including the 12.

Fix: Subtract 12 first: 3x = 18, then divide by 3 to get x = 6.

reversed_operands

Losing the negative sign

✗ For 5 − 2x = 17, writing 2x = 17 − 5 = 12 and then x = 6.

You dropped the minus in front of 2x. The term is −2x, so isolating it gives −2x = 12, not 2x = 12.

Fix: Keep the sign attached to the term: −2x = 12, then divide by −2 to get x = −6. Dividing by a negative flips the sign of the answer.

sign_error

Inverting the wrong operation

✗ For x/6 + 4 = 10, subtracting 4 gives x/6 = 6, then the student divides by 6 to get x = 1.

x was divided by 6, so its inverse is multiplication. Dividing again does the opposite of what's needed.

Fix: Multiply both sides by 6: x = 36. Match each operation to its true inverse — divide undoes multiply, multiply undoes divide.

wrong_operation

Practice: two-step equations

Peel each operation off from the outside in. Watch for negative coefficients and equations that put the variable on the right — the naive shortcut is often the trap.

Related skills

First, make sure of

Frequently asked questions

What exactly makes an equation "two-step"?

The variable is wrapped in exactly two operations — typically a multiplication or division plus an addition or subtraction. For example, 3x + 7 = 22 needs two inverse steps to isolate x: one to clear the +7 and one to clear the ×3.

Why do I subtract before I divide?

Because you undo operations in the reverse of the order they were applied. In 3x + 7, the x was multiplied by 3 first and had 7 added last. To take it apart you reverse that: remove the 7 first, then divide by 3. It is the order of operations run backwards.

How do I handle a negative in front of the variable, like 9 − 4x = 33?

Keep the sign attached to the term. Isolate −4x to get −4x = 24, then divide by −4. Because you are dividing by a negative, the answer's sign flips, giving x = −6. Forgetting that minus is one of the most common errors at this level.

Does it matter if the variable is on the right, like 40 = 3x − 8?

Not to the method — an equation reads the same both ways. Undo the last operation first: add 8 to both sides to get 48 = 3x, then divide by 3 for x = 16. Don't be tempted to divide 40 by 3 before clearing the −8.

How can I be sure my answer is right?

Substitute it back into the original equation and check both sides are equal. For x = 16 in 40 = 3x − 8: 3(16) − 8 = 48 − 8 = 40. If the two sides match, your solution is correct.