Find the k-value so that the given points have the given slope between them. A(k+2, -3), B(6, k) and m=$$ \frac35 $$

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:

\[
k = -\frac{3}{8}
\]

---

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.

Find the k-value so that the given points have the given slope between them. A(k+2, -3), B(6, k) and m=3/5.

This problem explores how to find the value of k for points A(k+2, -3) and B(6, k) so that the slope between them equals 3/5. Includes step-by-step calculations using the slope formula. Ideal for students in Grades 9-10 studying coordinate geometry and algebra.

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