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Dealing with fractions in algebra can be tricky, especially when they appear in equations. To make things easier, we need to focus on eliminating these fractions so the equation can be solved more straightforwardly. In algebra, fractions appear as parts of terms as seen in expressions like \( \frac{x}{5} - \frac{x}{3} = -8 \). Here, fractions divide the variable by different numbers, making direct operations a bit complicated.

Why do we want to remove them? Because removing fractions makes calculations simpler and easier to handle. This can be achieved by transforming each term of the equation into whole numbers. We use a technique called clearing fractions. This involves multiplying each term across the equation by a common factor, which we'll discuss next as the least common multiple. By doing so, every term can be rewritten without fractions, paving the path to solve algebraic equations efficiently.

When clearing fractions, the least common multiple (LCM) takes center stage. The LCM of numbers helps us avoid fractions by providing us with a common factor to multiply through the equation.

Think of the LCM as the smallest number that both denominators can divide into without leftovers. In our exercise, the denominators are 5 and 3. Finding the LCM for these numbers involves listing the multiples of each (like 5, 10, 15 for 5 and 3, 6, 9, 12, 15 for 3) and picking the smallest common one, which is 15 in this case.

Multiplying each term in the equation by 15 helps to "clear" the fractions:

• Multiply \( \frac{x}{5} \) by 15 to get \( 3x \)
• Multiply \( \frac{x}{3} \) by 15 to get \( 5x \)
• The constant \(-8\) becomes \(-8 \times 15 = -120\)

Voila, no more fractions! Now, the equation \(3x - 5x = -120\) emerges, ready for the final steps of solving.

After clearing fractions and achieving a fraction-free equation, the next journey is simplifying it. This step involves reshaping the equation into one that is straightforward to solve. Let’s review how to do this with our example.

In the equation \(3x - 5x = -120\), notice that the terms on the left can be combined as they both involve the variable \(x\). By combining them, you get \(-2x = -120\).

The final task here is solving for \(x\). This is a typical linear equation step where we isolate the variable by dividing both sides by the coefficient of \(x\), which is -2 in this case.

• Divide \(-2x\) by -2 to get \(x\)
• Correspondingly, divide \(-120\) by -2 to reach the value of \(x = 60\)

Through this process of simplifying each step, the solution of the variable becomes clear without unnecessary complex operations, which is the beauty of equation simplification.