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Three points are on the same line if the slope of the line through the first two points is the same as the slope of the line through the second two points. For what value of $c$ are the three points $(2,4)$, $(6,3)$, and $(-5,c)$ on the same line?

 Nov 12, 2017

Best Answer 

 #1
avatar+7352 
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the slope between two points   \(=\,\frac{\text{difference in y values}}{\text{difference in x values}}\)

 

the slope between  (2, 4)  and  (6, 3)   \(=\,\frac{3-4}{6-2}\)

 

the slope between  (2, 4)  and  (6, 3)   \(=\,-\frac{1}{4}\)

 

And we know that the slope between  (6, 3)  and  (-5, c)  also equals  -\(\frac14\) .  So we know

 

\(-\frac14\,=\,\frac{c-3}{-5-6}\\~\\ -\frac14\,=\,\frac{c-3}{-11}\)

                                              Multiply both sides of the equation by  -11 .

\(\frac{11}4\,=\,c-3\)

                                              Add  3  to both sides.

\(\frac{11}4+3\,=\,c \\~\\ c\,=\,\frac{23}{4}\,=\,5.75\)

 

Here is a graph to show that these points lie on the same line.

 Nov 12, 2017
 #1
avatar+7352 
+1
Best Answer

the slope between two points   \(=\,\frac{\text{difference in y values}}{\text{difference in x values}}\)

 

the slope between  (2, 4)  and  (6, 3)   \(=\,\frac{3-4}{6-2}\)

 

the slope between  (2, 4)  and  (6, 3)   \(=\,-\frac{1}{4}\)

 

And we know that the slope between  (6, 3)  and  (-5, c)  also equals  -\(\frac14\) .  So we know

 

\(-\frac14\,=\,\frac{c-3}{-5-6}\\~\\ -\frac14\,=\,\frac{c-3}{-11}\)

                                              Multiply both sides of the equation by  -11 .

\(\frac{11}4\,=\,c-3\)

                                              Add  3  to both sides.

\(\frac{11}4+3\,=\,c \\~\\ c\,=\,\frac{23}{4}\,=\,5.75\)

 

Here is a graph to show that these points lie on the same line.

hectictar Nov 12, 2017

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