If `y = (x +2)/( x +1) , y ne 1,` then x equals

Options

A. `(y +2)/( 1 - y)`
B. `( y +2)/(y -1)`
C. `(y -2)/(y +1)`
D. `(2-y)/(1 - y)`

Answer

To solve the equation \( y = \frac{x + 2}{x + 1} \) for \( x \), given that \( y \neq 1 \), we can follow these steps:

### Step 1: Multiply both sides by the denominator
We start with the equation:
\[ y = \frac{x + 2}{x + 1} \]
To eliminate the fraction, we multiply both sides by \( (x + 1) \):
\[ y(x + 1) = x + 2 \]

### Step 2: Distribute \( y \) on the left side
Distributing \( y \) gives us:
\[ yx + y = x + 2 \]

### Step 3: Rearrange the equation
Next, we want to isolate the terms involving \( x \) on one side. We can rearrange the equation:
\[ yx - x = 2 - y \]

### Step 4: Factor out \( x \)
Now, we can factor \( x \) out of the left side:
\[ x(y - 1) = 2 - y \]

### Step 5: Solve for \( x \)
To find \( x \), we divide both sides by \( (y - 1) \) (noting that \( y \neq 1 \)):
\[ x = \frac{2 - y}{y - 1} \]

### Final Result
Thus, the solution for \( x \) is:
\[ x = \frac{2 - y}{y - 1} \]