💡 What You Will Learn
When an equation has fractions with x in the numerator (like (2x−3)/5 = x/10), the best strategy is to eliminate the fractions first using either cross multiplication (when both sides are single fractions) or multiplying by the LCD. Then solve the resulting linear equation normally.
When you see x inside a fraction, your first goal is to get rid of the fraction. Once the fractions are gone, you have a simple linear equation you already know how to solve.
The GED problem: (2x−3)/5 = x/10 has one fraction on each side — perfect for cross multiplication!
One fraction equals another fraction. Use cross multiplication: multiply diagonally.
a/b = c/d → ad = bc
Multiple fractions plus a number. Use the LCD method: multiply every term by the LCD to clear all fractions.
Is it one fraction = one fraction? → Cross multiply.
Are there multiple terms? → LCD method.
Cross multiply OR multiply by the LCD. This gives you a regular linear equation.
Use inverse operations — collect x terms, then isolate x. Same as any two-step equation!
Plug your answer into the original equation to verify both sides are equal.
If a/b = c/d, then a × d = b × c
Multiply the numerator of the LEFT by the denominator of the RIGHT, and set it equal to the denominator of the LEFT times the numerator of the RIGHT.
Cross multiplication works ONLY when the equation is in the form fraction = fraction (one fraction on each side, nothing else). If there are other terms added outside the fractions, use the LCD method instead.
10(2x − 3) = 5 · x
20x − 30 = 5x
20x − 5x − 30 = 0
15x − 30 = 0
15x = 30
Divide by 15:
x = 30 ÷ 15 = 2
Right: 2 / 10 = 1/5
1/5 = 1/5 ✅
| Choice | What Went Wrong | Correct? |
|---|---|---|
| x = 2 | Cross multiply correctly → x = 2 ✅ | ✅ CORRECT |
| x = 3 | Sign error or arithmetic mistake ❌ | ❌ Wrong |
| x = 1/5 | Confused the answer with 1/5 (from checking step) ❌ | ❌ Trap! |
| x = 3/10 | Didn't cross multiply correctly ❌ | ❌ Wrong |
Find the smallest number that both denominators divide into evenly. Multiply every term on both sides by the LCD. This eliminates all fractions at once!
LCD = 10
10 × (2x−3)/5 = 10 × x/10
2(2x − 3) = 1 · x
4x − 6 = x
3x − 6 = 0 → 3x = 6 → x = 2
Equation is exactly: fraction = fraction (one fraction on each side, nothing else added)
Multiple fractions, or a fraction plus a whole number. More flexible overall!
30 = 6x
Left numerator × Right denominator = Left denominator × Right numerator.
(2x−3) × 10 = 5 × x. NOT (2x−3) × x = 5 × 10!
10(2x − 3) = 20x − 30, NOT 20x − 3. Multiply the 10 by BOTH terms inside the parentheses!
After distributing, get all x terms on one side and constants on the other before dividing.
Plug x = 2 into (2(2)−3)/5 = 1/5, and into 2/10 = 1/5. Equal! ✅ This catches arithmetic errors.
If you're not sure whether to cross multiply, find the LCD of all denominators and multiply every term by it — this always works and eliminates all fractions.
| Method | When to Use | Key Step |
|---|---|---|
| Cross Multiply | fraction = fraction only | a/b = c/d → ad = bc |
| LCD Method | any fraction equation | multiply ALL terms by LCD |