The ordered pairs below represent a relation between x and y.(-3,1), (-2,3), (-1,5), (0,7), (1,9), (2,11)Could this set of ordered pairs have been generated by a linear function?A. No, because the distance between consecutive y-values is different than the distance between consecutive x-valuesB. Yes, because the distance between consecutive x-values is constantC. Yes, because the relative difference between y-values and x-values is the same no matter which pairs of (x, y) values you use to calculate itD. No, because the y-values decrease and then increase

Notice that the difference between consecutive values of x is always 1, and the difference between consecutive values of y is always 2. This means that

\(\frac{x_{1}-x_{0}}{y_{1}-y_{0}} = 2\)

where \(x_0\) and \(x_1\) stands for two consecutive values of x, and the same goes to y.

Now, notice that this is the definition of the slope of the equation of a line. Then, a line can be characterized by this condition, and that is why C is the correct answer.

Anyway, you can check that this set of point is generated by a linear function in other way. Suppose that it can be done, and the linear function has the form

\(y=mx+n\).

From the pair (0,7) we deduce that \(7 =m*0+n\), then \(n=7\). Now, substituting the pair (-3,1) we get \(1=-3*m+7\) and as consequence \(m=2\). Thus, if the points are generated by a linear function, its expression would be \(y=2x+7\).

Now, substitute the different values of x and check that the same values of y are obtained.