To find the value of $k$ so that the points $A(k+2, -3)$ and $B(6, k)$ have a slope of $m = \frac{3}{5}$, we use the slope formula:

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

Given:

• Point $A$ with coordinates $(k+2, -3)$
• Point $B$ with coordinates $(6, k)$
• Slope $m = \frac{3}{5}$

Substitute these values into the slope formula:

$\frac{3}{5} = \frac{k - (-3)}{6 - (k + 2)}$

Simplify this equation:

1. $\frac{3}{5} = \frac{k + 3}{6 - k - 2}$
2. $\frac{3}{5} = \frac{k + 3}{4 - k}$

Now, cross-multiply to solve for $k$:

$3(4 - k) = 5(k + 3)$

Expanding both sides:

$12 - 3k = 5k + 15$

Combine like terms:

$12 - 15 = 5k + 3k$ $-3 = 8k$

Now, divide by 8:

$k = -\frac{3}{8}$

Therefore, the value of $k$ that makes the slope between points $A(k+2, -3)$ and $B(6, k)$ equal to $\frac{3}{5}$ is:

$k = -\frac{3}{8}$

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Would you like further explanation on any step? Here are some additional related questions you might consider:

1. What if the slope was different, say $\frac{2}{3}$? How would that change the result?
2. How would you find $k$ if one of the coordinates had an additional variable?
3. Can we verify the result by substituting back into the original slope equation?
4. How would you solve for $k$ if the slope was given as a negative fraction?
5. What other methods can you use to find $k$ in similar problems?

Tip: Always check your solution by substituting it back into the original equation to confirm it satisfies the conditions.