Solving Equations with Fractions on Both Sides

🧮 Example:

Solve:

(2x/3) − (1/6) = (x/2) + (1/3)

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🔍 Step 1: Identify the Denominators

The denominators are: 3, 6, 2, and 3

To simplify, we eliminate the fractions by multiplying all terms by the Least Common Multiple (LCM) of these numbers.

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✏️ Step 2: Find the LCM

Let’s list a few multiples:

Multiples of 2: 2, 4, 6, 8...

Multiples of 3: 3, 6, 9...

Multiples of 6: 6, 12, 18...

✅ The smallest common multiple is 6

So the LCM = 6

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✏️ Step 3: Multiply Every Term by 6

Now apply this to the equation:

6 × (2x/3) − 6 × (1/6) = 6 × (x/2) + 6 × (1/3)

Now simplify:

6 × (2x/3) = 4x

6 × (1/6) = 1

6 × (x/2) = 3x

6 × (1/3) = 2

Now rewrite the new equation:

4x − 1 = 3x + 2

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✏️ Step 4: Solve the New Equation

Now solve like a regular linear equation:

4x − 1 = 3x + 2

Subtract 3x from both sides:

x − 1 = 2

Add 1 to both sides:

x = 3

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✅ Final Answer:

x = 3

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💡 Quick Tip:

When dealing with equations that include fractions:

Always begin by finding the LCM of all denominators.

Multiply every term in the equation by the LCM to eliminate the fractions.

Then solve step-by-step like a regular equation.

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