Finding the missing variable in order to find slope.

[Unicorn0409]

The three points (3, -5), (-d + 2, 3) and (2d + 3, 2) lie on the same line. What is d?

 

[R.M.]

If all three points are on the same line, then what do you know about the slope between any two of them?

 

[Mr. Bland]

The slope between any two points on a line is always the same no matter which two points you consider. It is found by taking the "rise" (the difference in Y) and dividing it by the "run" (the distance in X). This division produces the same result proportionally for any pair of points, even if you reverse the order.

Given two points [MATH](x_1, y_1)[/MATH] and [MATH](x_2, y_2)[/MATH], the slope can be calculated like this:

[MATH]m = \frac{y_2 - y_1}{x_2 - x_1}[/MATH]​

Three points are given in the problem:

1. [MATH]\left(3, -5\right)[/MATH]
2. [MATH]\left(-d + 2, 3\right)[/MATH]
3. [MATH]\left(2d + 3, 2\right)[/MATH]

Consider what happens when we find the slope using points 1 and 2, then points 1 and 3:

[MATH]m = \frac{3 - (-5)}{(-d + 2) - 3} = \frac{8}{-d - 1}[/MATH]
[MATH]m = \frac{2 - (-5)}{(2d + 3) - 3} = \frac{7}{2d}[/MATH]​

As these are both solutions for [MATH]m[/MATH], we can take them as equal to one another:

[MATH]\frac{8}{-d - 1} = \frac{7}{2d}[/MATH]​

Now we have an equation where we can solve for [MATH]d[/MATH].