To determine the value of [tex]\( k \)[/tex] such that the points [tex]\((3, 2)\)[/tex], [tex]\((0, -4)\)[/tex], and [tex]\((-3, k)\)[/tex] are collinear, we need to ensure that the slopes between any two pairs of these points are equal.
Let's first calculate the slope between the points [tex]\((3, 2)\)[/tex] and [tex]\((0, -4)\)[/tex]:
The slope formula between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
For the points [tex]\((3, 2)\)[/tex] and [tex]\((0, -4)\)[/tex]:
[tex]\[
x_1 = 3, \quad y_1 = 2, \quad x_2 = 0, \quad y_2 = -4
\][/tex]
[tex]\[
\text{slope} = \frac{-4 - 2}{0 - 3} = \frac{-6}{-3} = 2
\][/tex]
Next, we calculate the slope between the points [tex]\((3, 2)\)[/tex] and [tex]\((-3, k)\)[/tex]:
Using the same slope formula:
[tex]\[
x_1 = 3, \quad y_1 = 2, \quad x_3 = -3, \quad y_3 = k
\][/tex]
[tex]\[
\text{slope} = \frac{k - 2}{-3 - 3} = \frac{k - 2}{-6}
\][/tex]
For the points to be collinear, the slope between [tex]\((3, 2)\)[/tex] and [tex]\((0, -4)\)[/tex] must be equal to the slope between [tex]\((3, 2)\)[/tex] and [tex]\((-3, k)\)[/tex]. Thus, we set these slopes equal:
[tex]\[
2 = \frac{k - 2}{-6}
\][/tex]
Now, solve for [tex]\( k \)[/tex]:
First, multiply both sides by [tex]\(-6\)[/tex] to clear the denominator:
[tex]\[
2 \times (-6) = k - 2
\][/tex]
[tex]\[
-12 = k - 2
\][/tex]
Next, solve for [tex]\( k \)[/tex] by adding 2 to both sides of the equation:
[tex]\[
-12 + 2 = k
\][/tex]
[tex]\[
k = -10
\][/tex]
Therefore, the value of [tex]\( k \)[/tex] that makes the points [tex]\((3, 2)\)[/tex], [tex]\((0, -4)\)[/tex], and [tex]\((-3, -10)\)[/tex] collinear is [tex]\(-10\)[/tex].
