# Given tha points (-6, 1), (1, t), and (10, 5), determine the value of t such that the three points are collinear.

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Note that points are collinear when they lie on the same line.

In order for the points (-6,1),(1,t) and (10,5) to be collinear, the slope between (-6,1) & (10,5) , (1,t) & (10,5) , and (-6,1) & (1,t) should all be the same.

To solve for t, the slope of (-6,1)and (10,5) should be determined.

m = (y2-y1) / (x2-x1) = (5-1) / (10- (-6)) = 4/16= 1/4

This means that the slope of the line where the three points lie is 1/4.

Then, substitute points (1,t) & (10,5) , and the slope 1/4 to the formula above.

m = (y2-y1) / (x2-x1)

1/4 = (5 - t) / (10 - 1)

1/4 = (5 - t)/ 9

Multiply both sides by 9.

9/4 = 5 - t

Subtract both sides by 5.

9/4 - 5 = -t

9/4 - 20/4 = -t

-11/4 = -t

11/4 = t

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To check, substitute the value of t to point (1,t).

Then, solve for the slope of (-6,1) and (1,11/4).

m = (11/4 - 1) / (1- (-6)) = (7/4) / 7 = 1/4

Also, solve for the slope of (1, 11/4) and (10,5).

m = (5 - 11/4) / (10 - 1) = (9/4) / 9 = 1/4

Since the the slopes between (-6,1) & (1,11/4), (1,11/4) & (10,5) , and (10,5) and (-6,1) are the same, hence the three points are collinear.

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**Answer: The value of t that makes (-6,1), (1,t) and (10,5) collinear is 11/4.**

since the points are collinear,

they all will satisfy the equation of a common line.

the slope of the line is given as : 5-1/10-(-6)

=4/16

= 4

using one point form of a line,

y-5 = 1/4 *(x -10)

4y-20=x-10

x- 4y + 10 = 0 is the equation of the required line.

now since (1 , t) should satisfy the equation

1 -4*t + 10=0

11-4t =0

t = 11/4