`(8x)/(3 (x-5))+ (2x)/(3x -15)=(50)/(3 (x-5))`
What value (s) of x satisfy the equation above ?

Options

A. 0
B. 5
C. No solution
D. Any value such that `x ne 5`

Answer

To solve the equation

\[ \frac{8x}{3(x-5)} + \frac{2x}{3x - 15} = \frac{50}{3(x-5)}, \]

we will follow these steps:

### Step 1: Simplify the second term
Notice that \(3x - 15\) can be factored as \(3(x - 5)\). Thus, we can rewrite the equation as:

\[ \frac{8x}{3(x-5)} + \frac{2x}{3(x-5)} = \frac{50}{3(x-5)}. \]

### Step 2: Combine the fractions on the left side
Now that both fractions on the left side have the same denominator, we can combine them:

\[ \frac{8x + 2x}{3(x-5)} = \frac{50}{3(x-5)}. \]

This simplifies to:

\[ \frac{10x}{3(x-5)} = \frac{50}{3(x-5)}. \]

### Step 3: Eliminate the denominators
Since the denominators are the same (and assuming \(x \neq 5\) to avoid division by zero), we can multiply both sides by \(3(x-5)\):

\[ 10x = 50. \]

### Step 4: Solve for \(x\)
Now, we can solve for \(x\):

\[ x = \frac{50}{10} = 5. \]

### Step 5: Check for validity
However, we must check if \(x = 5\) is a valid solution. Substituting \(x = 5\) into the original equation results in division by zero in the denominators \(3(x-5)\). Therefore, \(x = 5\) is not a valid solution.

### Conclusion
Since \(x = 5\) makes the equation undefined, there are no solutions to the equation.

### Final Answer
The equation has **no solution**.

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