The value of x that satisfies the equation `4/(x-3)+5/(x-5)=9/(x-13)` is

Options

A. 4
B. 3
C. 2
D. 1

Answer

To solve the equation \( \frac{4}{x-3} + \frac{5}{x-5} = \frac{9}{x-13} \), we will follow these steps:

### Step 1: Find a common denominator
The common denominator for the left side of the equation is \( (x-3)(x-5) \). The right side has a denominator of \( (x-13) \).

### Step 2: Rewrite the equation
We can rewrite the left side by multiplying each term by the common denominator:
\[ \frac{4}{x-3}(x-3)(x-5) + \frac{5}{x-5}(x-3)(x-5) = \frac{9}{x-13}(x-3)(x-5) \]
This simplifies to:
\[ 4(x-5) + 5(x-3) = 9 \cdot \frac{(x-3)(x-5)}{(x-13)} \]

### Step 3: Expand both sides
Expanding the left side:
\[ 4x - 20 + 5x - 15 = 9 \cdot \frac{(x-3)(x-5)}{(x-13)} \]
Combine like terms:
\[ 9x - 35 = 9 \cdot \frac{(x-3)(x-5)}{(x-13)} \]

### Step 4: Cross-multiply
Cross-multiply to eliminate the fraction:
\[ (9x - 35)(x - 13) = 9(x - 3)(x - 5) \]

### Step 5: Expand both sides again
Expanding the left side:
\[ 9x^2 - 117x - 35x + 455 = 9(x^2 - 8x + 15) \]
This simplifies to:
\[ 9x^2 - 152x + 455 = 9x^2 - 72x + 135 \]

### Step 6: Move all terms to one side
Subtract \( 9x^2 \) from both sides:
\[ -152x + 455 = -72x + 135 \]
Rearranging gives:
\[ -152x + 72x + 455 - 135 = 0 \]
This simplifies to:
\[ -80x + 320 = 0 \]

### Step 7: Solve for \( x \)
Rearranging gives:
\[ 80x = 320 \]
Dividing both sides by 80:
\[ x = \frac{320}{80} = 4 \]

### Final Answer
Thus, the value of \( x \) that satisfies the equation is \( x = 4 \).

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