Two-Step Equations: Combining Inverse Operations in the Right Order

What Makes an Equation “Two-Step”

A two-step equation usually has two operations applied to the variable. Your job is to undo those operations using inverse operations until the variable is alone.

The key idea is the reverse order of operations: if the variable is multiplied and then something is added, you undo the addition/subtraction first, then undo the multiplication/division.

• If the equation looks like ax + b = c, undo + b (or - b) first, then undo × a.
• If the equation looks like x/a + b = c, undo + b first, then undo ÷ a (by multiplying).

Keep your work clean: write each new line as a full equation, and make the same change to both sides.

Annotated Worked Examples (Line-by-Line)

Example 1: 2x + 3 = 11

Goal: isolate x. The variable term 2x has + 3 attached, so undo that first.

2x + 3 = 11          (start)
2x + 3 - 3 = 11 - 3  (subtract 3 from both sides)
2x = 8              (simplify)
2x / 2 = 8 / 2      (divide both sides by 2)
x = 4               (simplify)

Check: substitute x = 4: 2(4) + 3 = 8 + 3 = 11 ✓

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Example 2: 5x - 10 = 25

Notice: - 10 is subtraction. Undo it by adding 10 first.

5x - 10 = 25            (start)
5x - 10 + 10 = 25 + 10  (add 10 to both sides)
5x = 35                (simplify)
5x / 5 = 35 / 5         (divide both sides by 5)
x = 7                   (simplify)

Check: 5(7) - 10 = 35 - 10 = 25 ✓

Example 3: x/4 + 2 = 7

Notice: the variable is being divided by 4, then 2 is added. Undo + 2 first, then undo division by multiplying.

x/4 + 2 = 7          (start)
x/4 + 2 - 2 = 7 - 2  (subtract 2 from both sides)
x/4 = 5              (simplify)
4(x/4) = 5·4          (multiply both sides by 4)
x = 20               (simplify)

Check: 20/4 + 2 = 5 + 2 = 7 ✓

Keeping Work Readable (So Errors Don’t Sneak In)

• One change per line: it’s easier to spot mistakes.
• Write the operation you did: for example, “subtract 3 from both sides.”
• Don’t jump straight to the answer if it causes arithmetic slips (like forgetting a negative).
• Use parentheses when needed to keep negatives clear, e.g., -3x and -(x/2) are not the same as -x/2 in every context.

Common Pitfalls (and How to Avoid Them)

Pitfall 1: Distributing when it’s not needed

In 2x + 3 = 11, there is nothing to distribute because 2x is not written like 2(x + 3). If you start distributing anyway, you’ll create extra terms and make the problem harder.

Pitfall 2: Skipping steps that cause arithmetic errors

Going from 5x - 10 = 25 straight to x = 7 might be possible mentally, but it’s easy to mis-add and get 5x = 15 by mistake. Writing 5x = 35 clearly prevents that.

Pitfall 3: Mishandling negative values

When you add or subtract a negative, treat it carefully:

• x - (-4) becomes x + 4.
• x + (-4) is the same as x - 4.

Guided Practice (Partially Completed Steps)

Fill in the missing steps. Keep each line as a balanced equation.

1) 3x + 5 = 20

3x + 5 = 20
3x + 5 - ___ = 20 - ___
3x = ___
3x / ___ = ___ / ___
x = ___

2) 7x - 9 = 40

7x - 9 = 40
7x - 9 + ___ = 40 + ___
7x = ___
7x / 7 = ___ / 7
x = ___

3) x/6 + 3 = 9

x/6 + 3 = 9
x/6 + 3 - ___ = 9 - ___
x/6 = ___
6(x/6) = ___·6
x = ___

4) x/5 - 2 = 6

x/5 - 2 = 6
x/5 - 2 + ___ = 6 + ___
x/5 = ___
5(x/5) = ___·5
x = ___

Independent Practice

Solve each equation. Show clean, line-by-line work.

A. Build confidence (no negatives, no fractions)

• 4x + 7 = 31
• 9x - 18 = 45
• 6x + 2 = 50
• 8x - 3 = 29

B. Negative coefficients (watch the sign)

• -2x + 5 = 17
• -3x - 4 = 11
• -5x + 20 = -5
• -4x - 9 = -25

C. Simple fractions (variable divided by a number)

• x/3 + 4 = 10
• x/8 - 1 = 5
• x/2 + 7 = 1
• x/5 - 6 = -2

D. Mixed challenge (negatives and fractions together)

• x/4 - 3 = -1
• x/6 + 2 = -3
• -2x + 1 = -9
• -x/3 + 5 = 2

Quick Self-Check: Did You Undo in the Right Order?