Answer:
(-2,4), (-1,8), (0, 12), (1, 16) is a set of points in a straight line
Step-by-step explanation:
Points On A Line
If we are given a set of points (x1,y1),(x2,y2)(x3,y3),... they are part of a line if, between each pair of them, the slope is constant. The slope of a line, given two points, is
![\displaystyle m=\frac{y_2-y_1}{x_2-x_1}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20m%3D%5Cfrac%7By_2-y_1%7D%7Bx_2-x_1%7D)
We are given these sets of points
(3,6) (6, 12) (12, 19) (15, 13)
(-2,4), (-1,8), (0, 12), (1, 16)
(2,9), (1,7), (0, -14), (-1,20)
Let's try the first one
(3,6) (6, 12) (12, 19) (15, 13)
The first slope is
![\displaystyle m_1=\frac{12-6}{6-3}=2](https://tex.z-dn.net/?f=%5Cdisplaystyle%20m_1%3D%5Cfrac%7B12-6%7D%7B6-3%7D%3D2)
The next slope is
![\displaystyle m_2=\frac{19-12}{12-6}=\frac{7}{6}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20m_2%3D%5Cfrac%7B19-12%7D%7B12-6%7D%3D%5Cfrac%7B7%7D%7B6%7D)
Both values are different, so the set is not part of a line
Now for the second set
(-2,4), (-1,8), (0, 12), (1, 16)
Here are the slopes
![\displaystyle m_1=\frac{8-4}{-1+2}=4](https://tex.z-dn.net/?f=%5Cdisplaystyle%20m_1%3D%5Cfrac%7B8-4%7D%7B-1%2B2%7D%3D4)
![\displaystyle m_2=\frac{12-8}{0+1}=4](https://tex.z-dn.net/?f=%5Cdisplaystyle%20m_2%3D%5Cfrac%7B12-8%7D%7B0%2B1%7D%3D4)
![\displaystyle m_3=\frac{16-12}{1-0}=4](https://tex.z-dn.net/?f=%5Cdisplaystyle%20m_3%3D%5Cfrac%7B16-12%7D%7B1-0%7D%3D4)
All of them are equal, so these points lie in the same line
The third set of points results are
![\displaystyle m_1=\frac{7-9}{1-2}=2](https://tex.z-dn.net/?f=%5Cdisplaystyle%20m_1%3D%5Cfrac%7B7-9%7D%7B1-2%7D%3D2)
![\displaystyle m_2=\frac{-14-7}{0-1}=21](https://tex.z-dn.net/?f=%5Cdisplaystyle%20m_2%3D%5Cfrac%7B-14-7%7D%7B0-1%7D%3D21)
The slopes are different. This set is not part of a line