If ` (5)/( 2) - ((6)/( 5)) ( x - (15)/( 2)) = (- x)/( 5)` then what is the value of x ?

Options

A. (a) `(-23)/(2)`
B. (b) `(13)/(2)`
C. (c) `(-13)/(2)`
D. (d) `(23)/(2)`

Answer

To solve the equation \( \frac{5}{2} - \frac{6}{5}(x - \frac{15}{2}) = -\frac{x}{5} \), we will follow these steps:

### Step 1: Distribute the term on the left side
We start by distributing \( -\frac{6}{5} \) across \( (x - \frac{15}{2}) \):

\[ \frac{5}{2} - \frac{6}{5}x + \frac{6}{5} \cdot \frac{15}{2} = -\frac{x}{5} \]

### Step 2: Simplify the constant term
Now, we calculate \( \frac{6}{5} \cdot \frac{15}{2} \):

\[ \frac{6 \cdot 15}{5 \cdot 2} = \frac{90}{10} = 9 \]

So, the equation becomes:

\[ \frac{5}{2} - \frac{6}{5}x + 9 = -\frac{x}{5} \]

### Step 3: Combine like terms
Next, we combine \( \frac{5}{2} \) and \( 9 \):

\[ 9 = \frac{18}{2} \]

Thus,

\[ \frac{5}{2} + \frac{18}{2} = \frac{23}{2} \]

The equation now looks like this:

\[ \frac{23}{2} - \frac{6}{5}x = -\frac{x}{5} \]

### Step 4: Move all \( x \) terms to one side
We can add \( \frac{6}{5}x \) to both sides:

\[ \frac{23}{2} = -\frac{x}{5} + \frac{6}{5}x \]

This simplifies to:

\[ \frac{23}{2} = \frac{5x}{5} \]

### Step 5: Simplify the equation
This simplifies further to:

\[ \frac{23}{2} = x \]

### Step 6: Final answer
Thus, the value of \( x \) is:

\[ x = \frac{23}{2} \]