To solve for [tex]\( y \)[/tex] in the given hyperbola equation [tex]\( \frac{(x-3)^2}{9} - \frac{(y-4)^2}{4} = 1 \)[/tex] when [tex]\( x = -5 \)[/tex], follow these steps:
1. Substitute [tex]\( x = -5 \)[/tex] into the equation:
[tex]\[
\frac{((-5) - 3)^2}{9} - \frac{(y-4)^2}{4} = 1
\][/tex]
2. Simplify the expression:
[tex]\[
\frac{(-8)^2}{9} - \frac{(y-4)^2}{4} = 1
\][/tex]
[tex]\[
\frac{64}{9} - \frac{(y-4)^2}{4} = 1
\][/tex]
3. Isolate the term involving [tex]\( y \)[/tex]:
[tex]\[
\frac{64}{9} - 1 = \frac{(y-4)^2}{4}
\][/tex]
[tex]\[
\frac{64}{9} - \frac{9}{9} = \frac{64 - 9}{9} = \frac{55}{9}
\][/tex]
[tex]\[
\frac{55}{9} = \frac{(y-4)^2}{4}
\][/tex]
4. Multiply both sides by 4 to clear the fraction on the right-hand side:
[tex]\[
4 \cdot \frac{55}{9} = (y-4)^2
\][/tex]
[tex]\[
\frac{220}{9} = (y-4)^2
\][/tex]
5. Take the square root of each side to solve for [tex]\( y \)[/tex]:
[tex]\[
y - 4 = \pm \sqrt{\frac{220}{9}}
\][/tex]
[tex]\[
y - 4 = \pm \frac{\sqrt{220}}{3}
\][/tex]
6. Rewrite the expression to find both possible [tex]\( y \)[/tex] values:
[tex]\[
y = 4 + \frac{\sqrt{220}}{3} \quad \text{and} \quad y = 4 - \frac{\sqrt{220}}{3}
\][/tex]
7. Evaluate the square root and simplify, then round to the nearest integer:
[tex]\[
y_1 = 4 + \frac{\sqrt{220}}{3} \approx 4 + \frac{14.83}{3} \approx 4 + 4.94 \approx 9
\][/tex]
[tex]\[
y_2 = 4 - \frac{\sqrt{220}}{3} \approx 4 - \frac{14.83}{3} \approx 4 - 4.94 \approx -1
\][/tex]
Thus, for [tex]\( x = -5 \)[/tex], the [tex]\( y \)[/tex]-values to the nearest integer are:
[tex]\[
(9, -1)
\][/tex]
