Which of the following equations represents a linear relationship?

• A. xy = k
• B. y = mx + b
• C. y = ax^2 + bx + c
• D. y = sin(x)

Answer: (B)

The equation that represents a linear relationship is the one in the form y = mx + b, where m is the slope and b is the y-intercept. This format is characteristic of a linear equation, depicting a straight line when graphed on a coordinate plane. In this case, y depends linearly on x, meaning that for every unit increase in x, y changes by a constant amount given by the slope m.

In contrast, the other equations present different relationships. The equation xy = k represents a hyperbola, indicating a multiplicative relationship rather than a linear one. The quadratic equation y = ax^2 + bx + c describes a parabolic curve, illustrating a non-linear relationship where the change in y accelerates or decelerates based on the value of x. Lastly, y = sin(x) signifies a periodic function, showcasing a sine wave that oscillates between fixed values, which also does not depict a linear relationship.

Thus, the form of y = mx + b clearly indicates linearity through consistent change, making it the correct representation of a linear relationship among the options.