Which is an equation of a direct proportion?
A y = 1/6x + c
B y = 6/x
C y = 6x – 6
D y = 6x
The correct answer and explanation is:
The correct equation for a direct proportion is:
D) y = 6x
Explanation:
A direct proportion means that two quantities increase or decrease together at a constant rate. In mathematical terms, when y is directly proportional to x, it can be expressed as: y=kxy = kx
where k is a constant of proportionality.
Let’s examine each option:
- A) y = 1/6x + c: This equation represents a linear relationship, but not a direct proportion. The constant c suggests that there is an offset (a y-intercept), meaning that the relationship is not proportional, as the line does not pass through the origin.
- B) y = 6/x: This represents an inverse proportion, not a direct proportion. In an inverse proportion, as x increases, y decreases, and vice versa, so it cannot be classified as a direct proportion.
- C) y = 6x – 6: This is a linear equation, but like option A, it contains a constant term (-6) which represents a shift in the line. This means y is not directly proportional to x; it has a nonzero y-intercept.
- D) y = 6x: This is the equation of a direct proportion. The constant 6 represents the proportionality factor (or constant of proportionality). It means that for every unit increase in x, y will increase by 6 units, and vice versa. The line passes through the origin (0,0), which is a defining characteristic of a direct proportion.
Therefore, the correct answer is D. The equation y = 6x clearly demonstrates a direct relationship where y changes proportionally with x.