Given three points $A(3,8)$, $B(4,m)$, and $C(-1,4)$, which are collinear (lie on the same straight line), find the value of $m$.

1. Problem Description

Given three points $A(3,8)$, $B(4,m)$, and $C(-1,4)$, which are collinear (lie on the same straight line), find the value of $m$.

2. Solution Steps

If three points are collinear, then the slope between any two pairs of points must be the same.

We can calculate the slope between points A and C, and then use that slope to find the value of m using points A and B.

The slope between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

$slope = \frac{y_2 - y_1}{x_2 - x_1}$

First, calculate the slope between A(3,8) and C(-1,4):

$slope_{AC} = \frac{4 - 8}{-1 - 3} = \frac{-4}{-4} = 1$

Next, calculate the slope between A(3,8) and B(4,m):

$slope_{AB} = \frac{m - 8}{4 - 3} = \frac{m - 8}{1} = m - 8$

Since the points are collinear, $slope_{AC} = slope_{AB}$:

$1 = m - 8$

$m = 1 + 8$

$m = 9$

3. Final Answer

The value of $m$ is

9.