Line L contains the points given in the table below. If the slope of line m is 1/3 the slope of line L what are three points that could be on line m?
x 1 2 3
y -7 5 17
A.(1,-2),(2,10),(3,22)
B.(1,-7),(2,3),(3,1)
C.(1,-7),(2,2),(3,6)
D.(1,-7),(2,2),(3,22)
To find the slope of line m, we first need to find the slope of line L using the given points.
The slope of a line can be calculated using the formula:
slope = (change in y) / (change in x)
Let's calculate the slope of line L using the given points (1, -7), (2, 5), and (3, 17):
For the points (1, -7) and (2, 5):
slope = (5 - (-7)) / (2 - 1)
= 12 / 1
= 12
For the points (2, 5) and (3, 17):
slope = (17 - 5) / (3 - 2)
= 12 / 1
= 12
We can see that the slope of line L is 12.
Now, to find the slope of line m, which is 1/3 the slope of line L:
slope of line m = (1/3) * slope of line L
= (1/3) * 12
= 4
So, the slope of line m is 4.
Now, we can find three points that could be on line m using the slope of 4 and any x and y values. Let's substitute various x values into the equation of line m to find the corresponding y values:
For x = 1:
y = slope * x + y-intercept
= 4 * 1 + y-intercept
= 4 + y-intercept
So, one possible point on line m is (1, 4 + y-intercept).
Similarly, for x = 2:
y = slope * x + y-intercept
= 4 * 2 + y-intercept
= 8 + y-intercept
So, another possible point on line m is (2, 8 + y-intercept).
And for x = 3:
y = slope * x + y-intercept
= 4 * 3 + y-intercept
= 12 + y-intercept
So, a third possible point on line m is (3, 12 + y-intercept).
From the answer choices, we need to find three points that satisfy the above conditions.
Let's check each option:
A.(1,-2), (2,10), (3,22)
If we substitute these points into the equation of line m, we get:
(1,-2), (2,10), (3,22)
For x = 1: y = 4 + y-intercept -> 4 + y-intercept ≠ -2
For x = 2: y = 8 + y-intercept -> 8 + y-intercept ≠ 10
For x = 3: y = 12 + y-intercept -> 12 + y-intercept ≠ 22
Therefore, option A is not correct.
B.(1,-7), (2,3), (3,1)
For x = 1: y = 4 + y-intercept -> 4 + y-intercept ≠ -7
For x = 2: y = 8 + y-intercept -> 8 + y-intercept ≠ 3
For x = 3: y = 12 + y-intercept -> 12 + y-intercept ≠ 1
Therefore, option B is not correct.
C.(1,-7), (2,2), (3,6)
For x = 1: y = 4 + y-intercept -> 4 + y-intercept = -7
For x = 2: y = 8 + y-intercept -> 8 + y-intercept = 2
For x = 3: y = 12 + y-intercept -> 12 + y-intercept = 6
Therefore, option C is correct.
D.(1,-7), (2,2), (3,22)
For x = 1: y = 4 + y-intercept -> 4 + y-intercept = -7
For x = 2: y = 8 + y-intercept -> 8 + y-intercept = 2
For x = 3: y = 12 + y-intercept -> 12 + y-intercept ≠ 22
Therefore, option D is not correct.
In conclusion, the three points that could be on line m are (1, -7), (2, 2), and (3, 6). Therefore, the correct answer is option C.