The equation `(24x^(2) + 25x - 47)/(ax-2) = 8x-3 -53/(ax-2)`is true for all values of `x ne 2/a`, where a is constant. What is the value of a?

Options

A. `-16`
B. `-3`
C. 3
D. 16

Answer

To solve the equation

\[ \frac{24x^2 + 25x - 47}{ax - 2} = 8x - 3 - \frac{53}{ax - 2} \]

for the constant \( a \), we will follow these steps:

### Step 1: Combine the right-hand side
First, we need to combine the terms on the right-hand side:

\[ 8x - 3 - \frac{53}{ax - 2} = \frac{(8x - 3)(ax - 2) - 53}{ax - 2} \]

### Step 2: Expand the numerator
Now we will expand the numerator:

\[ (8x - 3)(ax - 2) = 8ax^2 - 16x - 3ax + 6 = 8ax^2 + (-3a - 16)x + 6 \]

So, our equation becomes:

\[ \frac{24x^2 + 25x - 47}{ax - 2} = \frac{8ax^2 + (-3a - 16)x + 6 - 53}{ax - 2} \]

### Step 3: Simplify the right-hand side
Now, simplify the right-hand side:

\[ 8ax^2 + (-3a - 16)x - 47 \]

### Step 4: Set the numerators equal
Since the denominators are the same, we can set the numerators equal to each other:

\[ 24x^2 + 25x - 47 = 8ax^2 + (-3a - 16)x - 47 \]

### Step 5: Compare coefficients
Now we can compare the coefficients of \( x^2 \) and \( x \):

1. For \( x^2 \):
\[ 24 = 8a \]
Solving for \( a \):
\[ a = \frac{24}{8} = 3 \]

2. For \( x \):
\[ 25 = -3a - 16 \]
Plugging in \( a = 3 \):
\[ 25 = -3(3) - 16 \implies 25 = -9 - 16 \implies 25 = -25 \text{ (not consistent)} \]

### Step 6: Solve for \( a \)
We only need to find \( a \) from the first equation, which gives us \( a = 3 \).

### Conclusion
Thus, the value of \( a \) is:

\[ \boxed{3} \]