To find the coordinates of the point that is [tex]\(\frac{1}{5}\)[/tex] of the way from [tex]\(A(-7, -4)\)[/tex] to [tex]\(B(3, 6)\)[/tex], we need to determine the difference in the [tex]\(x\)[/tex]-coordinates and the [tex]\(y\)[/tex]-coordinates between points [tex]\(A\)[/tex] and [tex]\(B\)[/tex], then scale these differences by [tex]\(\frac{1}{5}\)[/tex], and finally add these scaled differences to the coordinates of point [tex]\(A\)[/tex].
Here's the step-by-step solution:
1. Calculate the difference in the [tex]\(x\)[/tex]-coordinates:
[tex]\[
\Delta x = B_x - A_x = 3 - (-7) = 10
\][/tex]
2. Calculate the difference in the [tex]\(y\)[/tex]-coordinates:
[tex]\[
\Delta y = B_y - A_y = 6 - (-4) = 10
\][/tex]
3. Scale the differences by [tex]\(\frac{1}{5}\)[/tex]:
[tex]\[
\text{Scale in } x \text{-direction} = \frac{1}{5} \times \Delta x = \frac{1}{5} \times 10 = 2
\][/tex]
[tex]\[
\text{Scale in } y \text{-direction} = \frac{1}{5} \times \Delta y = \frac{1}{5} \times 10 = 2
\][/tex]
4. Add the scaled differences to the coordinates of point [tex]\(A\)[/tex]:
[tex]\[
\text{New } x \text{-coordinate} = A_x + \text{Scale in } x \text{-direction} = -7 + 2 = -5
\][/tex]
[tex]\[
\text{New } y \text{-coordinate} = A_y + \text{Scale in } y \text{-direction} = -4 + 2 = -2
\][/tex]
Thus, the coordinates of the point that is [tex]\(\frac{1}{5}\)[/tex] of the way from [tex]\(A(-7,\ -4)\)[/tex] to [tex]\(B(3,\ 6)\)[/tex] are [tex]\((-5, -2)\)[/tex].
The correct answer is:
[tex]\[ \boxed{(-5, -2)} \][/tex]
