A Line $$ A B $$ joing the points $$ A(3 M) $$ and $$ B(5,8)^{A} $$ (c) Find the equation of the line $$ A B $$.

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To find the equation of the line $$ AB $$ joining the points $$ A(3, M) $$ and $$ B(5, 8) $$, we can use the two-point form of the equation of a line.

The formula for the slope $$ m $$ of the line through two points $$ (x_1, y_1) $$ and $$ (x_2, y_2) $$ is given by:

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

In our case, the points are $$ A(3, M) $$ and $$ B(5, 8) $$. Thus, we have:

- $$ x_1 = 3 $$, $$ y_1 = M $$
- $$ x_2 = 5 $$, $$ y_2 = 8 $$

Now, we can calculate the slope $$ m $$:

$$
m = \frac{8 - M}{5 - 3} = \frac{8 - M}{2}
$$

Next, we can use the point-slope form of the equation of a line, which is:

$$
y - y_1 = m(x - x_1)
$$

Using point $$ A(3, M) $$:

$$
y - M = \frac{8 - M}{2}(x - 3)
$$

Now, we can simplify this equation:

1. Distribute the slope on the right side:

$$
y - M = \frac{8 - M}{2}x - \frac{(8 - M)}{2} \cdot 3
$$

2. Simplifying further:

$$
y - M = \frac{8 - M}{2}x - \frac{3(8 - M)}{2}
$$

3. Adding $$ M $$ to both sides:

$$
y = \frac{8 - M}{2}x - \frac{3(8 - M)}{2} + M
$$

4. To combine the constant terms, we can express $$ M $$ as $$ \frac{2M}{2} $$:

$$
y = \frac{8 - M}{2}x - \frac{3(8 - M)}{2} + \frac{2M}{2}
$$

5. Combine the constant terms:

$$
y = \frac{8 - M}{2}x + \frac{2M - 3(8 - M)}{2}
$$

6. Simplifying the constant term:

$$
2M - 3(8 - M) = 2M - 24 + 3M = 5M - 24
$$

Thus, the equation of the line $$ AB $$ becomes:

$$
y = \frac{8 - M}{2}x + \frac{5M - 24}{2}
$$

This is the equation of the line $$ AB $$ in slope-intercept form.
The equation of the line $$ AB $$ is $$ y = \frac{8 - M}{2}x + \frac{5M - 24}{2} $$.

A Line \( A B \) joing the points \( A(3 M) \) and \( B(5,8)^{A} \) (c) Find the equation of the line \( A B \).

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