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

[tex]\frac{x_{1}-x_{0}}{y_{1}-y_{0}} = 2[/tex]

where [tex]x_0[/tex] and [tex]x_1[/tex] 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

[tex]y=mx+n[/tex].

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

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