Collinear Coordinates

For points A, B and C to be colinear means that there is a linear equation $y = mx + c$ that satisfies all the coordinates.

First we make use of the known points, namely $A(-4, - 5)$ and $C(8, 19)$ . Calculating the gradient $m$ :

$m = \frac{rise}{run}$

$m = \frac{19 - -5}{8 - - 4}$

$m = \frac{24}{12}$

$m = 2$

Now the equation becomes $y = 2x + c$ . The next step is to calculate $c$ , which is equal to the y coordinate of the y-intercept. The y-intercept is the point in which our equation crosses the y-axis, namely when $x = 0$ .

To do so, we plug in one of the points to the current equation:

$y = 2x + c$

$19 = 2(8) + c$

$19 = 16 + c$

$c = 19 - 16$

$c = 3$

Now that we have our linear equation: $y = 2x + 3$ , we can substitute in our last point $B(p, 3p + 1)$ to solve for $p$ :

$y = 2x + 3$

$3p + 1 = 2(p) + 3$

$3p - 2p = 3 - 1$

$p = 2$

And thus we arrive at our answer, $p = 2$ .

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