To solve this question, we need to find the ratio in which the point [tex]\((3, y)\)[/tex] divides the line segment joining the points [tex]\((9,8)\)[/tex] and [tex]\((-4,-6)\)[/tex], and then use this to determine the value of [tex]\(y\)[/tex].
### Step-by-Step Solution:
1. Identify the given points:
- Let [tex]\(A(9,8)\)[/tex] be the point [tex]\((x_1, y_1)\)[/tex].
- Let [tex]\(B(-4,-6)\)[/tex] be the point [tex]\((x_2, y_2)\)[/tex].
- Let [tex]\(P(3, y)\)[/tex] be the point that divides the line segment.
2. Define the formula for the ratio:
If the point [tex]\(P(x, y)\)[/tex] divides the line segment [tex]\(AB\)[/tex] in the ratio [tex]\(k:1\)[/tex], we use the section formula which states:
[tex]\[
x = \frac{kx_1 + x_2}{k + 1}
\][/tex]
We are given that the x-coordinate of point [tex]\(P\)[/tex] is 3. Thus,
[tex]\[
3 = \frac{k \cdot 9 + (-4)}{k + 1}
\][/tex]
3. Solve for the ratio [tex]\(k\)[/tex]:
[tex]\[
3 = \frac{9k - 4}{k + 1}
\][/tex]
Multiply through by [tex]\(k + 1\)[/tex] to clear the denominator:
[tex]\[
3(k + 1) = 9k - 4
\][/tex]
[tex]\[
3k + 3 = 9k - 4
\][/tex]
Rearrange to solve for [tex]\(k\)[/tex]:
[tex]\[
3 + 4 = 9k - 3k
\][/tex]
[tex]\[
7 = 6k
\][/tex]
[tex]\[
k = \frac{7}{6} \approx 1.1667
\][/tex]
4. Use the ratio to find the y-coordinate [tex]\(y\)[/tex]:
The y-coordinate can be found using the section formula for the y-coordinates:
[tex]\[
y = \frac{ky_1 + y_2}{k + 1}
\][/tex]
Substituting [tex]\(k = \frac{7}{6}\)[/tex], [tex]\(y_1 = 8\)[/tex], and [tex]\(y_2 = -6\)[/tex]:
[tex]\[
y = \frac{\left(\frac{7}{6} \times 8\right) + (-6)}{\left(\frac{7}{6} + 1\right)}
\][/tex]
Simplify inside the numerator and the denominator:
[tex]\[
y = \frac{\left(\frac{56}{6}\right) - 6}{\left(\frac{7}{6} + \frac{6}{6}\right)}
\][/tex]
[tex]\[
y = \frac{\frac{56}{6} - \frac{36}{6}}{\frac{13}{6}}
\][/tex]
[tex]\[
y = \frac{\frac{20}{6}}{\frac{13}{6}}
\][/tex]
[tex]\[
y = \frac{20}{13} \approx 1.5385
\][/tex]
### Final Answer:
- The ratio in which the point [tex]\((3, y)\)[/tex] divides the line segment joining [tex]\((9,8)\)[/tex] and [tex]\((-4,-6)\)[/tex] is [tex]\(\frac{7}{6}\)[/tex] or approximately [tex]\(1.1667\)[/tex].
- The y-coordinate of the point is approximately [tex]\(1.5385\)[/tex].
