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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?

 Aug 16, 2017
edited by Guest  Aug 16, 2017

Best Answer 

 #1
avatar+1900 
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To solve this problem, first find the slope of the first two points.

 

\(m=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}\)

 

\(m=slope\)

 

\({y}_{2}=\)y-point in second point

 

\({y}_{1}=\)y-point in first point

 

\({x}_{2}=\)x-point in second point

 

\({x}_{1}=\)x-point in first point

 

\({y}_{2}=3\)

 

\({y}_{1}=4\)

 

\({x}_{2}=6\)

 

\({x}_{1}=2\)

 

\(m=\frac{3-4}{6-2}\)

 

\(m=\frac{-1}{6-2}\)

 

\(m=\frac{-1}{4}\)

 

\(m=-\frac{1}{4}\)

 

Next, do the same for the second and third points and solve for c.

 

\(m=-\frac{1}{4}\)

 

\({y}_{2}=c\)

 

\({y}_{1}=3\)

 

\({x}_{2}=-5\)

 

\({x}_{1}=6\)

 

\(-\frac{1}{4}=\frac{c-3}{-5-6}\)

 

\(-\frac{1}{4}=\frac{c-3}{-11}\)

 

\(-\frac{1}{4}\times-11=\frac{c-3}{-11}\times-11\)

 

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

 

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

 

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

 

\(\frac{11}{4}+\frac{12}{4}=c-3+3\)

 

\(\frac{11+12}{4}=c-3+3\)

 

\(\frac{23}{4}=c-3+3\)

 

\(5.75=c-3+3\)

 

\(5.75=c-0\)

 

\(5.75=c\)

 

\(c=5.75\)

.
 Aug 16, 2017
edited by gibsonj338  Aug 16, 2017
 #1
avatar+1900 
0
Best Answer

To solve this problem, first find the slope of the first two points.

 

\(m=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}\)

 

\(m=slope\)

 

\({y}_{2}=\)y-point in second point

 

\({y}_{1}=\)y-point in first point

 

\({x}_{2}=\)x-point in second point

 

\({x}_{1}=\)x-point in first point

 

\({y}_{2}=3\)

 

\({y}_{1}=4\)

 

\({x}_{2}=6\)

 

\({x}_{1}=2\)

 

\(m=\frac{3-4}{6-2}\)

 

\(m=\frac{-1}{6-2}\)

 

\(m=\frac{-1}{4}\)

 

\(m=-\frac{1}{4}\)

 

Next, do the same for the second and third points and solve for c.

 

\(m=-\frac{1}{4}\)

 

\({y}_{2}=c\)

 

\({y}_{1}=3\)

 

\({x}_{2}=-5\)

 

\({x}_{1}=6\)

 

\(-\frac{1}{4}=\frac{c-3}{-5-6}\)

 

\(-\frac{1}{4}=\frac{c-3}{-11}\)

 

\(-\frac{1}{4}\times-11=\frac{c-3}{-11}\times-11\)

 

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

 

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

 

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

 

\(\frac{11}{4}+\frac{12}{4}=c-3+3\)

 

\(\frac{11+12}{4}=c-3+3\)

 

\(\frac{23}{4}=c-3+3\)

 

\(5.75=c-3+3\)

 

\(5.75=c-0\)

 

\(5.75=c\)

 

\(c=5.75\)

gibsonj338 Aug 16, 2017
edited by gibsonj338  Aug 16, 2017

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