Quick Answer
The method is the same as any two-step equation — undo addition/subtraction first, then undo multiplication/division. With fractions, multiply both sides by the denominator to clear it. With negative coefficients, divide by the negative and watch the sign.
Why Fractions and Negatives Trip Students Up
The two-step method itself does not change when fractions or negatives appear. What changes is the arithmetic — and that is where most errors happen.
With fractions: students forget to multiply both sides by the denominator, or they only clear the fraction on one side.
With negative coefficients: students forget that dividing by a negative flips the sign of the result (and, for inequalities, flips the inequality sign too).
The fix is the same in both cases: slow down, write every step, and double-check the sign.
Solving with a Fractional Coefficient
When the variable has a fractional coefficient like (3/4)x, you have two options:
Option A — Multiply by the reciprocal: multiply both sides by 4/3 (the reciprocal of 3/4). This is the cleanest approach.
Option B — Multiply by the denominator first: multiply both sides by 4 to clear the fraction, then divide by 3.
Both give the same answer. Option A is usually faster for a single fraction.
Solving with a Negative Coefficient
When the coefficient is negative (e.g., −5x), divide both sides by −5 at the final step.
The result is just x = (number) ÷ (−5). Keep track of the sign:
• Positive ÷ negative = negative
• Negative ÷ negative = positive
For equations this is all you need. For inequalities, dividing by a negative also flips the inequality sign — but that is a separate topic.
Solving with a Variable in the Denominator
If the equation looks like x/4 + 3 = 7, the variable is divided by 4 (not multiplied by a fraction). The approach is the same:
Step 1: Undo the +3 by subtracting 3 from both sides → x/4 = 4.
Step 2: Undo the ÷4 by multiplying both sides by 4 → x = 16.
This is the most common form in Algebra 1 and is straightforward once you recognize that x/4 means x ÷ 4.
Worked Examples
Example 1
Solve: (3/4)x − 2 = 7
Solution
- Step 1: Add 2 to both sides to undo the −2.
- (3/4)x = 9
- Step 2: Multiply both sides by 4/3 (the reciprocal of 3/4).
- x = 9 · (4/3) = 36/3 = 12
- Check: (3/4)(12) − 2 = 9 − 2 = 7 ✓
Answer: x = 12
Example 2
Solve: −5x + 3 = 18
Solution
- Step 1: Subtract 3 from both sides.
- −5x = 15
- Step 2: Divide both sides by −5.
- x = 15 ÷ (−5) = −3
- Check: −5(−3) + 3 = 15 + 3 = 18 ✓
Answer: x = −3
Example 3
Solve: x/6 + 4 = −2
Solution
- Step 1: Subtract 4 from both sides.
- x/6 = −6
- Step 2: Multiply both sides by 6.
- x = −36
- Check: (−36)/6 + 4 = −6 + 4 = −2 ✓
Answer: x = −36
Example 4
Solve: (2/3)x + (1/2) = (5/6)
Solution
- Step 1: Subtract 1/2 from both sides.
- (2/3)x = 5/6 − 1/2 = 5/6 − 3/6 = 2/6 = 1/3
- Step 2: Multiply both sides by 3/2 (reciprocal of 2/3).
- x = (1/3) · (3/2) = 3/6 = 1/2
- Check: (2/3)(1/2) + 1/2 = 1/3 + 1/2 = 2/6 + 3/6 = 5/6 ✓
Answer: x = 1/2
Common Mistakes
Multiplying only the term with the fraction, not both sides. If you multiply by 4 to clear x/4, you must multiply every term on both sides by 4.
Forgetting the sign when dividing by a negative. −5x = 15 → x = −3, not x = 3.
Treating (3/4)x as 3/(4x). The coefficient 3/4 multiplies x; it does not put x in the denominator.
Skipping the check step. With fractions and negatives, arithmetic errors are common — always substitute your answer back.
Practice Problems
Solve: (1/2)x + 5 = 9
Hint: Subtract 5 first, then multiply by 2.
Solve: −3x − 7 = 11
Hint: Add 7 first, then divide by −3.
Solve: x/5 − 4 = −6
Hint: Add 4 first, then multiply by 5.
Solve: (2/5)x + 3 = 7
Hint: Subtract 3, then multiply by 5/2.
Solve: −(1/4)x + 6 = 2
Hint: Subtract 6, then multiply by −4.
Frequently Asked Questions
How do you solve a two-step equation with a fraction?
Use the same two-step method: undo addition or subtraction first, then undo the fraction. If the variable has a fractional coefficient like (3/4)x, multiply both sides by the reciprocal (4/3) to isolate x. If the variable is in the denominator like x/4, multiply both sides by 4.
How do you solve a two-step equation with a negative coefficient?
Follow the same two steps: undo addition or subtraction first, then divide both sides by the negative coefficient. Keep careful track of the sign — positive ÷ negative = negative, and negative ÷ negative = positive.
Do you flip the inequality sign when solving a two-step equation with a negative?
No — equations do not have inequality signs to flip. The flip rule applies only to inequalities (< > ≤ ≥) when you multiply or divide by a negative. For equations, you simply divide by the negative coefficient and track the sign of the result.
What is the reciprocal method for clearing fractions?
If the variable has a fractional coefficient a/b, multiply both sides by b/a (the reciprocal). This is equivalent to dividing by a/b and is usually the fastest way to isolate the variable when a single fraction is involved.
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