Final answer:
After implicit differentiation of the function 5xy + 2x - 12 = 0, and finding the slope at x = 1, the equation of the line tangent to the curve at that point is y = -5x + 5.
Step-by-step explanation:
To find an equation of the line tangent to the graph of the equation 5xy + 2x - 12 = 0 at the point where x = 1, we must first find the slope of the tangent line at that point. We do this by differentiating the given equation implicitly with respect to x to find the derivative dy/dx, which represents the slope of the tangent line at any point along the curve.
After differentiating, we evaluate the derivative at x = 1 to get the slope. Then we can use the point-slope form y - y1 = m(x - x1), where m is the slope and (x1, y1) is the given point, to find the tangent line's equation.
Implicit differentiation of 5xy + 2x - 12 = 0 gives us 5y + 5x(dy/dx) + 2 = 0. Evaluating at x = 1, we find the slope (dy/dx) and the y-coordinate of the curve. Finally, we get the equation of the tangent line y = -5x + 5, which fits option (b).