To find the [tex]\( y \)[/tex]-coordinate of the point that divides the line segment [tex]\(\overline{XY}\)[/tex] with endpoints [tex]\( X(-10, -1) \)[/tex] and [tex]\( Y(5, 15) \)[/tex] in a 5:3 ratio, follow these steps:
1. Identify the given values:
- [tex]\( a = 5 \)[/tex]
- [tex]\( b = 3 \)[/tex]
- [tex]\( y_1 = -1 \)[/tex] (the [tex]\( y \)[/tex]-coordinate of point [tex]\( X \)[/tex])
- [tex]\( y_2 = 15 \)[/tex] (the [tex]\( y \)[/tex]-coordinate of point [tex]\( Y \)[/tex])
2. Use the formula for the [tex]\( y \)[/tex]-coordinate of the point dividing the line segment in the given ratio:
[tex]\[
y = \left(\frac{a}{a + b}\right)(y_2 - y_1) + y_1
\][/tex]
3. Substitute the given values into the formula:
[tex]\[
y = \left(\frac{5}{5 + 3}\right)(15 - (-1)) + (-1)
\][/tex]
4. Simplify inside the parentheses:
[tex]\[
y = \left(\frac{5}{8}\right)(15 + 1) + (-1)
\][/tex]
[tex]\[
y = \left(\frac{5}{8}\right)(16) + (-1)
\][/tex]
5. Calculate the fraction multiplication:
[tex]\[
y = \left(\frac{5 \times 16}{8}\right) + (-1)
\][/tex]
[tex]\[
y = 10 + (-1)
\][/tex]
6. Finalize the calculation:
[tex]\[
y = 9
\][/tex]
Therefore, the [tex]\( y \)[/tex]-coordinate of the point that divides the line segment [tex]\(\overline{XY} \)[/tex] into a [tex]\( 5:3 \)[/tex] ratio is [tex]\( \boxed{9} \)[/tex].
