To find the points through which a line with a given slope passes, we can use the point-slope form of a linear equation:
y - y1 = m(x - x1)
Where m is the slope and (x1, y1) is a point on the line.
In this case, the slope is -5/3. Let's use the first set of points, (12, 13) and (17, 10), and substitute into the point-slope form:
y - 13 = (-5/3)(x - 12)
y - 13 = (-5/3)x + 20
y = (-5/3)x + 33
Let's check if these points satisfy the equation:
For (12, 13):
13 = (-5/3)(12) + 33
13 = -20 + 33
13 = 13 --> satisfied
For (17, 10):
10 = (-5/3)(17) + 33
10 = -85/3 + 99/3
10 = 14/3 --> not satisfied
The line does not pass through the points (12, 13) and (17, 10).
Let's move on to the second set of points, (16, 15) and (13, 10):
y - 15 = (-5/3)(x - 16)
y - 15 = (-5/3)x + 80/3
y = (-5/3)x + 145/3
Let's check if these points satisfy the equation:
For (16, 15):
15 = (-5/3)(16) + 145/3
15 = -80/3 + 145/3
15 = 65/3 --> not satisfied
For (13, 10):
10 = (-5/3)(13) + 145/3
10 = -65/3 + 145/3
10 = 80/3 --> not satisfied
The line does not pass through the points (16, 15) and (13, 10).
Lastly, let's try the third set of points, (0, 7) and (3, 10):
y - 7 = (-5/3)(x - 0)
y - 7 = (-5/3)x + 0
y = (-5/3)x + 7
Let's check if these points satisfy the equation:
For (0, 7):
7 = (-5/3)(0) + 7
7 = 0 + 7
7 = 7 --> satisfied
For (3, 10):
10 = (-5/3)(3) + 7
10 = -15/3 + 21/3
10 = 6/3
10 = 2*3 --> not satisfied
The line does not pass through the points (0, 7) and (3, 10).
Lastly, let's try the last set of points, (11, 13) and (8, 18):
y - 13 = (-5/3)(x - 11)
y - 13 = (-5/3)x + 55/3
y = (-5/3)x + 94/3
Let's check if these points satisfy the equation:
For (11, 13):
13 = (-5/3)(11) + 94/3
13 = -55/3 + 94/3
13 = 39/3 --> not satisfied
For (8, 18):
18 = (-5/3)(8) + 94/3
18 = -40/3 + 94/3
18 = 54/3
18 = 2*3 --> not satisfied
The line does not pass through the points (11, 13) and (8, 18).
Therefore, the line with a slope of -5/3 does not pass through any of the given sets of points.
Bot GPT 3.5 
Killer999 
Explain Bot