If line l in the x-y plane contains the point (-2,1),(4,5) and (5,t), what is the value of t?
Using the two-point form of a line:
(t-5)/(5-4) = (t-1)/(5+2)
t-5 = (t-1)/7
7t-35 = t-1
6t = 34
t = 17/3
You can see this by noting the slope of the line is 2/3.
So, as x increases by 1 from 4 to 5, y increases by 2/3 from 5 to 17/3.
To find the value of t, we need to determine the equation of the line that passes through the points (-2, 1), (4, 5), and (5, t).
First, let's find the slope of the line using the formula:
slope (m) = (y2 - y1) / (x2 - x1)
Taking the first two points (-2, 1) and (4, 5):
m = (5 - 1) / (4 - (-2))
= 4 / 6
= 2 / 3
Now, we have the slope (m) of the line.
Next, we'll use the point-slope form of a linear equation to find the equation of the line:
y - y1 = m(x - x1)
Taking the point (5, t) and substituting the coordinates, we have:
y - 5 = (2/3)(x - 5)
Expanding the equation, we get:
y - 5 = (2/3)x - (2/3)(5)
y - 5 = (2/3)x - (10/3)
Now, let's substitute the x-coordinate of the point (-2, 1) into the equation and solve for y:
1 - 5 = (2/3)(-2) - (10/3)
-4 = (-4/3) - (10/3)
-4 = -14/3
To find y, multiply both sides of the equation by 3:
-12 = -14
Since the equation has a contradiction (-12 = -14), it indicates that the given point (5, t) does not lie on the line passing through the points (-2, 1) and (4, 5).
Therefore, there is no unique value of t that satisfies the problem conditions.