The slope of the tangent line to the curve x+y=xy at the point (2,2) is

Options

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

Answer

To find the slope of the tangent line to the curve \(x + y = xy\) at the point \((2, 2)\), we will use implicit differentiation. Here’s a step-by-step solution:

### Step 1: Differentiate the equation implicitly
We start with the equation of the curve:
\[ x + y = xy \]
Now, we differentiate both sides with respect to \(x\):
\[ \frac{d}{dx}(x) + \frac{d}{dx}(y) = \frac{d}{dx}(xy) \]
Using the derivatives:
\[ 1 + \frac{dy}{dx} = y + x\frac{dy}{dx} \]

### Step 2: Rearrange the equation
Now, we will rearrange the equation to isolate \(\frac{dy}{dx}\):
\[ 1 + \frac{dy}{dx} = y + x\frac{dy}{dx} \]
Subtract \(x\frac{dy}{dx}\) from both sides:
\[ 1 + \frac{dy}{dx} - x\frac{dy}{dx} = y \]
Combine the \(\frac{dy}{dx}\) terms:
\[ 1 = y - \frac{dy}{dx}(x - 1) \]
Now, isolate \(\frac{dy}{dx}\):
\[ \frac{dy}{dx}(x - 1) = y - 1 \]
Thus, we have:
\[ \frac{dy}{dx} = \frac{y - 1}{x - 1} \]

### Step 3: Substitute the point (2, 2)
Now we will substitute the point \((2, 2)\) into the derivative:
\[ \frac{dy}{dx} = \frac{2 - 1}{2 - 1} = \frac{1}{1} = 1 \]

### Step 4: Conclusion
The slope of the tangent line to the curve \(x + y = xy\) at the point \((2, 2)\) is:
\[ \boxed{1} \]