$$ \frac{\left(x-1\right)^2}{3^2} $$+$$ \frac{\left(y-5\right)^2}{4^2} $$=1 solve for x

The given equation is:

\[
\frac{(x-1)^2}{3^2} + \frac{(y-5)^2}{4^2} = 1
\]

This is the equation of an ellipse in standard form:

\[
\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1
\]

Where $(h, k)$ is the center of the ellipse, and $a$ and $b$ are the semi-major and semi-minor axes, respectively. 

In your equation:
- $h = 1$
- $k = 5$
- $a = 3$
- $b = 4$

To solve for $x$, we can rearrange the equation:

\[
\frac{(x-1)^2}{3^2} = 1 - \frac{(y-5)^2}{4^2}
\]

Simplify:

\[
\frac{(x-1)^2}{9} = 1 - \frac{(y-5)^2}{16}
\]

Multiply both sides by 9:

\[
(x-1)^2 = 9\left(1 - \frac{(y-5)^2}{16}\right)
\]

Expand and simplify:

\[
(x-1)^2 = 9 - \frac{9(y-5)^2}{16}
\]

Take the square root of both sides:

\[
x - 1 = \pm \sqrt{9 - \frac{9(y-5)^2}{16}}
\]

Finally, solve for $x$:

\[
x = 1 \pm \sqrt{9 - \frac{9(y-5)^2}{16}}
\]

Thus, the solution for $x$ is:

\[
x = 1 \pm \sqrt{9 - \frac{9(y-5)^2}{16}}
\]

Would you like a detailed explanation on how to analyze this expression further, or do you have any specific questions? 

Here are some related questions you might consider:
1. What is the geometric interpretation of the given ellipse?
2. How do the values of $a$ and $b$ affect the shape of the ellipse?
3. How can you determine the range of values for $y$ in this context?
4. What is the effect of shifting the center of the ellipse on the solution for $x$?
5. How would you solve for $y$ if given a specific value for $x$?

**Tip:** When solving equations involving ellipses, understanding the relationship between the ellipse's geometry and the algebraic expression can simplify the solution process.

Solve the ellipse equation $$ \frac{(x-1)^2}{9} + \frac{(y-5)^2}{16} = 1 $$ for x, step-by-step. Learn about the geometric interpretation of ellipses and the relationship between the semi-major and semi-minor axes. Suitable for Grades 10-12 students studying coordinate geometry.

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