If `y=(x-2)/(x+1),yne1, then `x` equals

Options

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

Answer

To solve the equation \( y = \frac{x - 2}{x + 1} \) under the condition that \( y \neq 1 \), we will follow these steps:

### Step 1: Cross Multiply
Start with the equation:
\[ y = \frac{x - 2}{x + 1} \]
Cross-multiply to eliminate the fraction:
\[ y(x + 1) = x - 2 \]

### Step 2: Expand the Left Side
Distribute \( y \) on the left side:
\[ yx + y = x - 2 \]

### Step 3: Rearrange the Equation
Rearrange the equation to isolate terms involving \( x \):
\[ yx - x = -2 - y \]

### Step 4: Factor Out \( x \)
Factor out \( x \) from the left side:
\[ x(y - 1) = -2 - y \]

### Step 5: Solve for \( x \)
Now, solve for \( x \):
\[ x = \frac{-2 - y}{y - 1} \]

### Step 6: Simplify the Expression
We can rewrite the expression:
\[ x = \frac{-(y + 2)}{y - 1} \]

### Step 7: Consider the Condition \( y \neq 1 \)
Since \( y \neq 1 \), we can use this expression for \( x \) without any issues.

### Final Expression
Thus, the value of \( x \) in terms of \( y \) is:
\[ x = \frac{-(y + 2)}{y - 1} \]