box-o-sand/math-replay/algebra/solving-for-a-variable.tex

\documentclass{article}
\begin{document}

Solve for $l$:
\begin{eqnarray*}
-2lmn + 8m - 3n + 9 & = & -5m + 3n - 7 \\
-2lmn + 8m - 3n + 9 - 9 & = & -5m + 3n - 7 - 9 \\
-2lmn + 8m - 3n & = & -5m + 3n - 16 \\
-2lmn + 8m - 3n + 3n & = & -5m + 3n - 16 + 3n \\
-2lmn + 8m & = & -5m + 3n - 16 + 3n \\
-2lmn + 8m - 8m & = & -5m + 3n - 16 + 3n - 8m \\
-2lmn & = & -5m + 3n - 16 + 3n - 8m \\
\frac{-2lmn}{-2} & = & \frac{-5m + 3n - 16 + 3n - 8m}{-2} \\
lmn & = & \frac{-5m + 3n - 16 + 3n - 8m}{-2} \\
\frac{lmn}{mn} & = & \frac{\frac{-5m + 3n - 16 + 3n - 8m}{-2}}{mn} \\
l & = & \frac{\frac{-5m + 3n - 16 + 3n - 8m}{-2}}{mn} \\
l & = & \frac{\frac{-5m - 16 + 6n - 8m}{-2}}{mn} \\
l & = & \frac{\frac{-16 + 6n - 13m}{-2}}{mn} \\
l & = & \frac{8 - 3n - \frac{13m}{-2}}{mn} \\
\end{eqnarray*}

\begin{eqnarray*}
-2lmn + 8m - 3n + 9 & = & -5m + 3n - 7 \\
-2lmn + 8m - 3n + 9 - 9 & = & -5m + 3n - 7 - 9 \\
-2lmn + 8m - 3n & = & -5m + 3n + 2 \\
-2lmn + 8m - 3n + 3n & = & -5m + 3n + 2 + 3n \\
-2lmn + 8m & = & -5m + 6n + 2 \\
-2lmn + 8m - 8m & = & -5m + 6n + 2 - 8m \\
-2lmn & = & -13m + 6n + 2 \\
\frac{-2lmn}{-2} & = & \frac{-13m + 6n + 2}{-2} \\
lmn & = & \frac{-13m}{-2} + \frac{6n}{-2} + \frac{2}{-2} \\
lmn & = & \frac{-13m}{-2} + \frac{6n}{-2} - 1 \\
lmn & = & \frac{-13m}{-2} + \frac{3n}{-1} - 1 \\
lmn & = & \frac{-13m}{-2} - 3n - 1 \\
\frac{lmn}{mn} & = & \frac{\frac{-13m}{-2} - 3n - 1}{mn} \\
l & = & \frac{\frac{-13m}{-2} - 3n - 1}{mn} \\
\end{eqnarray*}

\begin{eqnarray*}
-2lmn + 8m - 3n + 9 & = & -5m + 3n - 7 \\
-9 + -2lmn + 8m - 3n + 9 & = & -9 + -5m + 3n - 7 \\
-2lmn + 8m - 3n & = & -9 + -5m + 3n - 7 \\
-2lmn + 8m - 3n & = & -5m + 3n - 7 - 9 \\
-2lmn + 8m - 3n & = & -5m + 3n + -7 + -9 \\
-2lmn + 8m - 3n & = & -5m + 3n + -16 \\
-2lmn + 8m - 3n + 3n & = & -5m + 3n + -16 + 3n \\
-2lmn + 8m & = & -5m + -16 + 6n \\
-2lmn + 8m - 8m & = & -5m + -16 + 6n - 8m \\
-2lmn & = & -13m + 6n + -16 \\
\frac{-2lmn}{-2mn} & = & \frac{-13m + 6n + -16}{-2mn} \\
l & = & \frac{-13m + 6n + -16}{-2mn} \\
l & = & \frac{13m - 6n + 16}{2mn} \\
\end{eqnarray*}

Solve for $x$:
\begin{eqnarray*}
-2xy + 5xz - 5x + 10 & = & -6y - 2 \\
-2xy + 5xz - 5x + 10 - 10 & = & -6y - 2 - 10 \\
-2xy + 5xz - 5x & = & -6y + -2 + -10 \\
-2xy + 5xz - 5x & = & -6y + -12 \\
-2xy + 5xz - 5xz - 5x & = & -6y - 12 - 5xz \\
-2xy - 5x & = & -6y - 12 - 5xz \\
-2xy + 2xy - 5x & = & -6y - 12 - 5xz + 2xy \\
-5x & = & -6y - 12 - 5xz + 2xy \\
\end{eqnarray*}

\end{document}