find the value of x so (x,-3) is collinear to (6,7) and (-2,-5). Show all work.
You want (x,-3) to be on the same straight line as the other two points.
Figure out the equation for that straight line. The slope is [7 - (-5)/[6 - (-2)] = 3/2
The line equation is y = (3/2)x + b
The value of b is given by
7 = (3/2)6 + b = 9 + b
Therefore b = -2 and the line equation is
y = (3/2)x - 2
Now if the point (x,-3) is going to be on that line, we must have
-3 = (3/2)x -2
3/2 x = -1
x = -2/3
OR
their slopes have to be equal
(-3-7)/(x-6) = (-5-7)/(-2-6)
-10/(x-6) = -12/-8
-12x + 72 = 80
-12x = 8
x = -8/12 = -2/3
To determine whether two points are collinear, we can use the slope formula. The slope between two points can be calculated using the formula:
m = (y2 - y1) / (x2 - x1)
Let's calculate the slopes between the given points:
For points (x, -3) and (6, 7):
m1 = (7 - (-3)) / (6 - x)
For points (x, -3) and (-2, -5):
m2 = (-5 - (-3)) / (-2 - x)
Since the points are collinear, the slopes between them should be the same. So, we can set m1 equal to m2 and solve for x:
(7 - (-3)) / (6 - x) = (-5 - (-3)) / (-2 - x)
Simplifying this equation steps by steps:
(7 + 3) / (6 - x) = (-5 + 3) / (-2 - x)
10 / (6 - x) = -2 / (-2 - x)
Now, let's cross multiply and solve for x by getting rid of the fractions:
10 * (-2 - x) = -2 * (6 - x)
-20 - 10x = -12 + 2x
To solve for x, let's combine like terms:
-10x - 2x = -12 + 20
-12x = 8
Divide both sides by -12 to isolate the variable:
x = 8 / -12
x = -2/3
Therefore, the value of x that makes the point (x, -3) collinear with (6, 7) and (-2, -5) is x = -2/3.