To find the [tex]$y$[/tex] value for the point [tex]$Y$[/tex] that is located [tex]\(\frac{1}{5}\)[/tex] of the distance from point [tex]\(X\)[/tex] to point [tex]\(Z\)[/tex], follow these steps:
1. Identify the coordinates of points [tex]\(X\)[/tex] and [tex]\(Z\)[/tex].
- Point [tex]\(X\)[/tex] is [tex]\((2, -6)\)[/tex].
- Point [tex]\(Z\)[/tex] is [tex]\((0, 5)\)[/tex].
2. Determine the change in the [tex]$x$[/tex] and [tex]$y$[/tex] coordinates between the points [tex]\(X\)[/tex] and [tex]\(Z\)[/tex].
- Change in [tex]$x$[/tex] (denoted as [tex]\(\Delta x\)[/tex]): [tex]\(0 - 2 = -2\)[/tex].
- Change in [tex]$y$[/tex] (denoted as [tex]\(\Delta y\)[/tex]): [tex]\(5 - (-6) = 5 + 6 = 11\)[/tex].
3. Calculate the [tex]$y$[/tex] coordinate for point [tex]$Y$[/tex] that is [tex]\(\frac{1}{5}\)[/tex] of the way from point [tex]\(X\)[/tex] to point [tex]\(Z\)[/tex].
- We use the fractional distance [tex]\(\frac{1}{5}\)[/tex] to find the proportionate change in the [tex]$y$[/tex] coordinate.
- Proportional change in [tex]$y$[/tex]: [tex]\(\frac{1}{5} \times \Delta y = \frac{1}{5} \times 11 = \frac{11}{5} = 2.2\)[/tex].
4. Calculate the [tex]$y$[/tex] coordinate of point [tex]$Y$[/tex]:
- Start from the [tex]$y$[/tex] coordinate of point [tex]\(X\)[/tex], which is [tex]\(-6\)[/tex].
- Add the proportional change in [tex]$y$[/tex]: [tex]\(-6 + 2.2 = -3.8\)[/tex].
Thus, the [tex]$y$[/tex] value for point [tex]\(Y\)[/tex] is [tex]\(-3.8\)[/tex].
Answer: [tex]\(-3.8\)[/tex].
