Final answer:
To find the equation of the tangent line to the curve x²−7xy=8 at the point (1, -1), we use implicit differentiation and the point-slope formula. The equation of the tangent line is y = (9/7)x - (16/7).
Step-by-step explanation:
To find the equation of the tangent line to the curve, we will use implicit differentiation. Starting with the equation: x²−7xy=8
First, we differentiate both sides of the equation with respect to x:
2x - 7x(dy/dx) - 7y = 0
Next, we need to find the derivative dy/dx. Rearranging the equation, we get:
dy/dx = (2x - 7y) / (7x)
Now, we substitute the coordinates of the given point (1, -1) into the derivative equation to find the slope at that point:
dy/dx = (2(1) - 7(-1)) / (7(1)) = 9/7
Therefore, the slope of the tangent line at the point (1, -1) is 9/7. Using the point-slope form of a linear equation, we can write the equation of the tangent line as: y - y1 = m(x - x1), where m is the slope and (x1, y1) is the given point. Plugging in the values, we get:
y - (-1) = (9/7)(x - 1)
Simplifying the equation gives:
y = (9/7)x - (16/7)