$$ \frac{2}{7x}+\frac14=\frac{53}{14x}-\frac13 $$

We are given the equation:

\[
\frac{2}{7x} + \frac{1}{4} = \frac{53}{14x} - \frac{1}{3}
\]

### Step 1: Eliminate the fractions by finding the least common denominator (LCD)

The denominators involved are $7x$, $4$, $14x$, and $3$. The least common denominator (LCD) would be the least common multiple of $7x$, $4$, and $3$, which is $84x$.

### Step 2: Multiply through by the LCD (84x)

Multiply each term by $84x$ to eliminate the fractions:

\[
84x \cdot \left(\frac{2}{7x} + \frac{1}{4}\right) = 84x \cdot \left(\frac{53}{14x} - \frac{1}{3}\right)
\]

Simplifying each term:

- $ 84x \cdot \frac{2}{7x} = 12 $
- $ 84x \cdot \frac{1}{4} = 21x $
- $ 84x \cdot \frac{53}{14x} = 318 $
- $ 84x \cdot \frac{1}{3} = 28x $

Thus, the equation becomes:

\[
12 + 21x = 318 - 28x
\]

### Step 3: Solve for $x$

Now, we'll move all the terms involving $x$ to one side and the constants to the other side:

\[
21x + 28x = 318 - 12
\]

\[
49x = 306
\]

### Step 4: Isolate $x$

Divide both sides by 49:

\[
x = \frac{306}{49}
\]

So, the solution is:

\[
x = \frac{306}{49} \quad \text{or approximately} \quad x \approx 6.24
\]

Would you like more details on any step, or do you have any questions?

---

Here are 5 related questions to explore this concept further:

1. How do you solve equations with multiple fractions?
2. What happens when the variable is in the denominator of a fraction?
3. How can you verify the solution to an equation involving fractions?
4. What are some common pitfalls when solving equations with fractions?
5. How do you factor the least common denominator in more complex equations?

**Tip:** Always check your solution by plugging the value of $x$ back into the original equation to ensure it satisfies the equation!

$$ \frac{2}{7x} + \frac{1}{4} = \frac{53}{14x} - \frac{1}{3} $$

This guide walks through solving the rational equation $ \frac{2}{7x} + \frac{1}{4} = \frac{53}{14x} - \frac{1}{3} $, step-by-step. Key concepts include the Least Common Denominator (LCD) and simplifying terms. Perfect for students in Grades 8-10 learning how to solve rational equations.

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