To find the value of $k$ so that the slope $m = \frac{3}{2}$ between points $A(k+1, -2)$ and $B(4, k)$, we can use the slope formula:

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

For points $A(k+1, -2)$ and $B(4, k)$, we have:

• $x_1 = k+1$
• $y_1 = -2$
• $x_2 = 4$
• $y_2 = k$

Substitute these values into the slope formula and set it equal to $\frac{3}{2}$:

$\frac{k - (-2)}{4 - (k+1)} = \frac{3}{2}$

Simplify this equation:

1. Rewrite the equation with subtraction: $\frac{k + 2}{4 - k - 1} = \frac{3}{2}$

2. Combine terms in the denominator: $\frac{k + 2}{3 - k} = \frac{3}{2}$

3. Cross multiply to solve for $k$: $2(k + 2) = 3(3 - k)$

4. Expand both sides: $2k + 4 = 9 - 3k$

5. Combine like terms: $5k = 5$

6. Solve for $k$: $k = 1$

Thus, the value of $k$ is $\mathbf{1}$.

Would you like further details on this process or have any questions?

Here are some related questions:

1. How is the slope formula derived, and when is it useful?
2. What happens if the points are vertical, and how does it affect the slope?
3. Can slope values indicate specific orientations of a line?
4. How does one find the slope if given coordinates are in a different form?
5. Why is cross-multiplying effective in solving slope-related equations?

Tip: When working with slopes, always check if your points result in a division by zero, as this would mean an undefined slope.