Why isn't \\ or \newline working?

Question

I can't figure out why neither of these commands (\\ or \newline) will work to start a new line for all of my equations.

\usepackage{amsmath}
\usepackage[utf8]{inputenc}


\begin{document}
\maketitle % showing both title and author, plus and automatic date
Consider the conic section
\begin{equation}

Ax^2+Bxy+Cy^2+Dx+Ey+F=0.

\end{equation}
If $B\neq0$, then to eliminate the xy term, we rotate the axes through an acute angle $\theta$ where 
\begin{equation}

\cot(2\theta)=\frac{A-C}{B}.

\end{equation}
In the equation
\begin{equation}
11x^2-24xy+4y^2+20=0,
\end{equation}
$A=11$, $B=-24$, $C=4$, $D=0$, $E=0$, and $F=20$. We will rotate the xy-axis     through an angle where
\begin{equation}
\cot(2\theta)=\frac{11-4}{-24}=\frac{-7}{24}.
\end{equation}
We will replace x and y with 
\begin{equation}
x=X\cos\theta-Y\sin\theta 
y=X\sin\theta+Y\cos\theta.
\end{equation}
We use the multiple angle formulas for sine and cosine that come from the   addition formulas for sine and cosine
\begin{equation}
\cos(2\theta)=\cos^2(\theta)-\sin^2(\theta) and 
\sin(2\theta)=2\sin(\theta)\cos(\theta).
\end{equation}

Answer 1

Is this what you're looking for

We will replace x and y with

\begin{align*}
x=&X\cos\theta-Y\sin\theta \\
y=&X\sin\theta+Y\cos\theta.
\end{align*}

It is better to use align environment. If you want these equations to be numbered :

We will replace x and y with

\begin{align}
x=&X\cos\theta-Y\sin\theta \\
y=&X\sin\theta+Y\cos\theta.
\end{align}

(Edit) If you want only one of the equation to be numbered :

We will replace x and y with

\begin{align}
x=&X\cos\theta-Y\sin\theta \nonumber\\
y=&X\sin\theta+Y\cos\theta.
\end{align}

Answer 2

Let me transform my comment to the answer:

Environment equation is intended for one line equation. For multi line equations you should use amsmath or mathtools (which is extended version of amsmath˙ package) environments like gather, align, multline, split to be numbered some of them. For details, read documentation for packages for amsmath or mathtools, or see LaTeX/Mathematics.

In example below is corrected your MWE, where is demonstrated use of split and align environment (from whole MWE is shown only part with use split and align)

Of course, select of appropriate amsmath environment is left to you.