Finding the Value of k from Slope and Points

The question asks us to find the value of 'k' for which the line joining the points (k, 4) and (-3, -2) has a slope of \(\frac{1}{2}\).

Understanding the Slope Formula

The slope of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula:

\[m = \frac{y_2 - y_1}{x_2 - x_1}\]

Applying the Formula to the Given Points

We are given the following information:

• Point 1: \((x_1, y_1) = (k, 4)\)
• Point 2: \((x_2, y_2) = (-3, -2)\)
• Slope: \(m = \frac{1}{2}\)

Substitute these values into the slope formula:

\[\frac{1}{2} = \frac{-2 - 4}{-3 - k}\]

Solving for the Value of k

Now, we need to solve this equation for 'k'.

First, simplify the numerator:

\[\frac{1}{2} = \frac{-6}{-3 - k}\]

Next, perform cross-multiplication:

\[1 \times (-3 - k) = 2 \times (-6)\]

\[-3 - k = -12\]

To isolate '-k', add 3 to both sides of the equation:

\[-k = -12 + 3\]

\[-k = -9\]

Finally, multiply both sides by -1 to find the value of k:

\[k = 9\]

Verifying the Solution

If k=9, the points are (9, 4) and (-3, -2). Let's calculate the slope:

\[m = \frac{-2 - 4}{-3 - 9} = \frac{-6}{-12} = \frac{6}{12} = \frac{1}{2}\]

This matches the given slope, confirming our value of k=9 is correct.

The value of k is 9.