A one-step equation is an equation you can solve in a single move: undo whatever operation is being done to the variable, and you have your answer. The core idea is using inverse (opposite) operations. Addition undoes subtraction, subtraction undoes addition, multiplication undoes division, and division undoes multiplication. Once you know which inverse to use, every one-step equation follows the same pattern.
The Golden Rule: Do the Same Thing to Both Sides
An equation is a balance. The left side equals the right side, and to keep it that way, anything you do to one side you must also do to the other. If you subtract 5 from the left, subtract 5 from the right. If you divide the left by 3, divide the right by 3. This single rule is the foundation for every example below.
Solving Addition and Subtraction Equations
When a number is added to the variable, subtract that number from both sides. When a number is subtracted from the variable, add it to both sides.
Example 1: x + 7 = 12
The 7 is being added to x, so subtract 7 from both sides:
- x + 7 − 7 = 12 − 7
- x = 5
Example 2: x − 4 = 9
The 4 is being subtracted from x, so add 4 to both sides:
- x − 4 + 4 = 9 + 4
- x = 13
A common mistake is using the wrong direction. If the equation has “minus 4,” some students subtract 4 again instead of adding it. Remember: you want to cancel the operation, not repeat it.
Solving Multiplication and Division Equations
When a number is multiplied by the variable, divide both sides by that number. When the variable is divided by a number, multiply both sides by that number.
Example 3: 3x = 18
The 3 is multiplied by x, so divide both sides by 3:
- 3x ÷ 3 = 18 ÷ 3
- x = 6
Example 4: x / 5 = 4
The x is being divided by 5, so multiply both sides by 5:
- (x / 5) × 5 = 4 × 5
- x = 20
A frequent error here is using subtraction instead of division. If you see 3x = 18 and subtract 3 from both sides, you get 3x − 3 = 15, which doesn’t isolate x at all. The 3 and x are connected by multiplication, so the inverse must be division.
Equations with Negative Numbers
Negative numbers follow the same rules. Just be careful with the signs.
Example 5: x + (−3) = 10, which is the same as x − 3 = 10
- x − 3 + 3 = 10 + 3
- x = 13
Example 6: −2x = 14
Divide both sides by −2:
- −2x ÷ (−2) = 14 ÷ (−2)
- x = −7
When you divide or multiply both sides by a negative number, the sign of the answer flips. In Example 6, a positive 14 divided by −2 gives −7.
Equations with Fractions
When the variable has a fractional coefficient (a fraction multiplied by it), multiply both sides by the reciprocal of that fraction. The reciprocal is just the fraction flipped: the numerator and denominator swap places.
Example 7: (2/3)x = 10
The reciprocal of 2/3 is 3/2. Multiply both sides by 3/2:
- (3/2) × (2/3)x = (3/2) × 10
- x = 30/2
- x = 15
This works because (3/2) × (2/3) equals 1, leaving x by itself on the left side. You could also divide both sides by 2/3, which is mathematically identical to multiplying by 3/2, but most students find the reciprocal method easier to execute without mistakes.
How to Check Your Answer
After solving, plug your answer back into the original equation and see if both sides are equal. This takes only a few seconds and catches arithmetic errors before they cost you points.
Checking Example 3: The original equation was 3x = 18, and you found x = 6.
- Substitute: 3(6) = 18
- 18 = 18 ✓
Checking Example 7: The original equation was (2/3)x = 10, and you found x = 15.
- Substitute: (2/3)(15) = 30/3 = 10
- 10 = 10 ✓
If the two sides don’t match, go back and look at which inverse operation you used. The most likely culprit is either picking the wrong inverse or making a sign error with negative numbers.
Quick Reference
- Equation has addition (x + a = b): Subtract a from both sides.
- Equation has subtraction (x − a = b): Add a to both sides.
- Equation has multiplication (ax = b): Divide both sides by a.
- Equation has division (x/a = b): Multiply both sides by a.
- Equation has a fractional coefficient ((a/b)x = c): Multiply both sides by the reciprocal (b/a).
Every one-step equation comes down to one question: what is being done to x, and what is the opposite? Answer that, apply the operation to both sides, and you have your solution.