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Question 530663: show that the three points (-3,4) (3,2) (6,1) lie in the same line
Answer by oberobic(2304)   (Show Source):
You can put this solution on YOUR website! The three points are: (-3,4) (3,2) (6,1).
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Use the first two points to define the slope of the line.
Then use the third point to define 'b' such that the line goes through the third point.
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m = (4-2) / (-3-3) = 2/-6 = -1/3
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y = -1/3x + b
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Use the third point (6,1) to define 'b'
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1 = (-1/3)(6) + b
1 = -2 + b
b = 3
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y = -1/3x +3
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Of course, the next question that could be asked is whether the defined line is the only line that contains all 3 points.
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Recall the three points are: (-3,4) (3,2) (6,1).
This time let's use (3,2) (6,1) to define the line and (-3,4) to find 'b'.
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m = (2-1)/(3-6) = 1/-3 = -1/3.
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We know immediately that with the same slope, the lines are parallel.
Given they're parallel and that they go through the same points, then they have to lie on top of one another.
Visually, they are the same line. But we can continue this example, to find the full equation.
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y = -1/3*x + b
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Use (-3,4) to find b.
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4 = -1/3*-3 + b
4 = 1 + b
b = 3
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y = -1/3*x + 3, which is the same equation as found above.
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Remember, this question asked if the 3 specific points lie on the same line. They do.
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However, you should realize any 3 points may not fit on the same line. There is no general law that any 3 points line on a straight line.
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Consider the points: (0,0), (1,1), and (0,1). They do not lie on the same line.
The points (0,0) and (1,1) line on the familiar line y=x, which has a slope = 1.
The line connecting (0,0) and (0,1) is a vertical line that has an undefined slope and does not go through (1,1).
The line connecting (0,1) and (1,1) is a horizontal line with slope = 0 and does not go through (0,0).
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Our intuition may be that if 3 lines lie on the same line, there is only one equation that fits the line to the three points.
But I leave it to you to prove or disprove that intuition.
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