If points ( , )A 3 12- , ( , )B 7 6 and ( , )C x 9 are collinear, then the value of x is ......

To find the value of x, we need to determine if points A, B, and C are collinear.

Collinear points are points that lie on the same straight line.

To check if points A, B, and C are collinear, we can use the slope formula. The slope between two points (x1, y1) and (x2, y2) is given by the formula:

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

Let's calculate the slope between points A and B:

slope_AB = (6 - 12) / (7 - 3) = -6 / 4 = -3/2

Now, let's calculate the slope between points B and C:

slope_BC = (9 - 6) / (x - 7) = 3 / (x - 7)

Since points A, B, and C are collinear, the slopes between AB and BC should be equal.

Therefore, we can set up the equation:

-3/2 = 3 / (x - 7)

To solve for x, we can cross-multiply:

-3(x - 7) = 2 * 3

Simplifying the equation:

-3x + 21 = 6

Subtracting 21 from both sides:

-3x = -15

Dividing both sides by -3:

x = 5

Therefore, the value of x is 5.