One-Step Equations: Using Inverse Operations to Isolate the Variable

What “inverse operations” do in one-step equations

A one-step equation has the variable in just one operation. Your goal is to isolate the variable (get it alone) by using the inverse operation—the operation that “undoes” what is happening to the variable.

• Addition ↔ Subtraction
• Multiplication ↔ Division

The key rule: whatever you do to one side of the equation, you must do to the other side. This keeps the two sides equal.

A consistent solving template (use this every time)

1. Identify the operation on the variable. Ask: “What is being done to x?”
2. Apply the inverse operation to both sides. Write it as a clear step.
3. Simplify. Compute the number side.
4. Check. Substitute your answer back into the original equation to verify equality.

Template in a quick layout

1) Identify operation on x: ______________________  ( +, −, ×, ÷ ? )
2) Apply inverse to both sides: __________________
3) Simplify: x = ______
4) Check in original: LHS = ____ , RHS = ____ (should match)

Type A: Addition equations (x + a = b)

If the equation adds a number to x, undo it by subtracting that number from both sides.

Example: x + 7 = 12

1. Identify operation on x: + 7
2. Apply inverse to both sides: subtract 7

x + 7 = 12
x + 7 − 7 = 12 − 7
x = 5

Check:

Original: x + 7 = 12
Substitute x = 5: 5 + 7 = 12 → 12 = 12 ✓

Practice set A (addition)

• x + 3 = 11
• x + 14 = 20
• x + 9 = 4
• x + 1.5 = 6
• x + 2/3 = 5/3

Type B: Subtraction equations (x − a = b)

If the equation subtracts a number from x, undo it by adding that number to both sides.

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Example: x − 5 = 9

1. Identify operation on x: − 5
2. Apply inverse to both sides: add 5

x − 5 = 9
x − 5 + 5 = 9 + 5
x = 14

Check:

Original: x − 5 = 9
Substitute x = 14: 14 − 5 = 9 → 9 = 9 ✓

Practice set B (subtraction)

• x − 8 = 2
• x − 6 = −1
• x − 0.4 = 3.1
• x − 7/5 = 3/5
• x − 12 = 30

Type C: Multiplication equations (ax = b)

If x is multiplied by a number, undo it by dividing both sides by that number.

Example: 4x = 20

1. Identify operation on x: × 4
2. Apply inverse to both sides: divide by 4

4x = 20
4x ÷ 4 = 20 ÷ 4
x = 5

Check:

Original: 4x = 20
Substitute x = 5: 4(5) = 20 → 20 = 20 ✓

Practice set C (multiplication)

• 7x = 49
• −3x = 18
• 0.5x = 6
• (2/3)x = 10
• 12x = −36

Type D: Division equations (x/a = b)

If x is divided by a number, undo it by multiplying both sides by that number.

Example: x/3 = 6

1. Identify operation on x: ÷ 3
2. Apply inverse to both sides: multiply by 3

x/3 = 6
(x/3) × 3 = 6 × 3
x = 18

Check:

Original: x/3 = 6
Substitute x = 18: 18/3 = 6 → 6 = 6 ✓

Practice set D (division)

• x/5 = 7
• x/(−4) = 3
• x/0.2 = 9
• x/(3/2) = 8
• x/12 = −2

Mixed practice set (build fluency)

Use the same template each time: identify → inverse on both sides → simplify → check.

• x + 10 = 3
• x − 9 = 15
• 6x = 54
• x/7 = 4
• −2x = −16
• x + 0.75 = 2.25
• x − (1/3) = 5/3
• x/(-6) = -5
• (1/4)x = 9
• x/2.5 = 6

Common mistakes (and how to avoid them)

1) Reversing signs incorrectly

What happens: You see x − 5 = 9 and subtract 5 again, getting x = 4.

Fix: Match the inverse operation to what is on x. If it’s − 5, the inverse is + 5.

2) Dividing when you should subtract (or subtracting when you should divide)

What happens: For x + 7 = 12, you divide by 7 because you notice the 7.

Fix: Don’t focus on the number; focus on the operation. Ask: “Is 7 being added, subtracted, multiplied, or divided?” Then choose the inverse.

3) Forgetting to apply the operation to both sides

What happens: You write x + 7 = 12 then jump to x = 5 without showing 12 − 7, or you only subtract on the left side.

Fix: Always write the balancing step explicitly:

x + 7 = 12
x + 7 − 7 = 12 − 7

This makes it harder to accidentally change only one side and break equality.