%html %head %script{:type => "text/x-mathjax-config"} MathJax.Hub.Config({tex2jax: {inlineMath: [['$','$'], ['\\(','\\)']]}}); %script{:type => "text/javascript", :src => "http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"} :css .solution { border-top: 1px dotted #222; } %body %h2 Solve for $l$: #solve-for-l-01.solution :escaped $$ \begin{eqnarray} -2lmn + 8m - 3n + 9 & = & -5m + 3n - 7 \\ -2lmn + 8m - 3n + 9 - 9 & = & -5m + 3n - 7 - 9 \\ -2lmn + 8m - 3n & = & -5m + 3n - 16 \\ -2lmn + 8m - 3n + 3n & = & -5m + 3n - 16 + 3n \\ -2lmn + 8m & = & -5m + 3n - 16 + 3n \\ -2lmn + 8m - 8m & = & -5m + 3n - 16 + 3n - 8m \\ -2lmn & = & -5m + 3n - 16 + 3n - 8m \\ \frac{-2lmn}{-2} & = & \frac{-5m + 3n - 16 + 3n - 8m}{-2} \\ lmn & = & \frac{-5m + 3n - 16 + 3n - 8m}{-2} \\ \frac{lmn}{mn} & = & \frac{\frac{-5m + 3n - 16 + 3n - 8m}{-2}}{mn} \\ l & = & \frac{\frac{-5m + 3n - 16 + 3n - 8m}{-2}}{mn} \\ l & = & \frac{\frac{-5m - 16 + 6n - 8m}{-2}}{mn} \\ l & = & \frac{\frac{-16 + 6n - 13m}{-2}}{mn} \\ l & = & \frac{8 - 3n - \frac{13m}{-2}}{mn} \\ \end{eqnarray} $$ #solve-for-l-02.solution :escaped $$ \begin{eqnarray} -2lmn + 8m - 3n + 9 & = & -5m + 3n - 7 \\ -2lmn + 8m - 3n + 9 - 9 & = & -5m + 3n - 7 - 9 \\ -2lmn + 8m - 3n & = & -5m + 3n + 2 \\ -2lmn + 8m - 3n + 3n & = & -5m + 3n + 2 + 3n \\ -2lmn + 8m & = & -5m + 6n + 2 \\ -2lmn + 8m - 8m & = & -5m + 6n + 2 - 8m \\ -2lmn & = & -13m + 6n + 2 \\ \frac{-2lmn}{-2} & = & \frac{-13m + 6n + 2}{-2} \\ lmn & = & \frac{-13m}{-2} + \frac{6n}{-2} + \frac{2}{-2} \\ lmn & = & \frac{-13m}{-2} + \frac{6n}{-2} - 1 \\ lmn & = & \frac{-13m}{-2} + \frac{3n}{-1} - 1 \\ lmn & = & \frac{-13m}{-2} - 3n - 1 \\ \frac{lmn}{mn} & = & \frac{\frac{-13m}{-2} - 3n - 1}{mn} \\ l & = & \frac{\frac{-13m}{-2} - 3n - 1}{mn} \\ \end{eqnarray} $$ #solve-for-l-03.solution :escaped $$ \begin{eqnarray} -2lmn + 8m - 3n + 9 & = & -5m + 3n - 7 \\ -9 + -2lmn + 8m - 3n + 9 & = & -9 + -5m + 3n - 7 \\ -2lmn + 8m - 3n & = & -9 + -5m + 3n - 7 \\ -2lmn + 8m - 3n & = & -5m + 3n - 7 - 9 \\ -2lmn + 8m - 3n & = & -5m + 3n + -7 + -9 \\ -2lmn + 8m - 3n & = & -5m + 3n + -16 \\ -2lmn + 8m - 3n + 3n & = & -5m + 3n + -16 + 3n \\ -2lmn + 8m & = & -5m + -16 + 6n \\ -2lmn + 8m - 8m & = & -5m + -16 + 6n - 8m \\ -2lmn & = & -13m + 6n + -16 \\ \frac{-2lmn}{-2mn} & = & \frac{-13m + 6n + -16}{-2mn} \\ l & = & \frac{-13m + 6n + -16}{-2mn} \\ l & = & \frac{13m - 6n + 16}{2mn} \\ \end{eqnarray} $$ %h2 Solve for $x$: #solve-for-x-01.solution :escaped $$ \begin{eqnarray} -2xy + 5xz - 5x + 10 & = & -6y - 2 \\ -2xy + 5xz - 5x + 10 - 10 & = & -6y - 2 - 10 \\ -2xy + 5xz - 5x & = & -6y + -2 + -10 \\ -2xy + 5xz - 5x & = & -6y + -12 \\ -2xy + 5xz - 5xz - 5x & = & -6y - 12 - 5xz \\ -2xy - 5x & = & -6y - 12 - 5xz \\ -2xy + 2xy - 5x & = & -6y - 12 - 5xz + 2xy \\ -5x & = & -6y - 12 - 5xz + 2xy \\ -5x \over -5 & = & -6y - 12 - 5xz + 2xy \over -5x \\ x & = & -6y - 12 - 5xz + 2xy \over -5x \\ x & = & \frac{-6y}{-5x} - \frac{12}{-5x} - \frac{5xz}{-5x} + \frac{2xy}{-5x} \\ \end{eqnarray} $$ #solve-for-x-02.solution :escaped $$ \begin{eqnarray} -2xy + 5xz - 5x + 10 & = & -6y - 2 \\ -2xy + 5xz - 5x + 10 - 10 & = & -6y - 2 - 10 \\ -2xy + 5xz - 5x & = & -6y + -2 + -10 \\ -2xy + 5xz - 5x & = & -6y - 12 \\ -2xy + 5xz - 5x \over 5z & = & -6y - 12 \over 5z \\ -2xy + x - 5x & = & -6y - 12 \over 5z \\ -2xy + x + -5x & = & -6y - 12 \over 5z \\ -2xy + -4x & = & -6y - 12 \over 5z \\ -2xy - 4x & = & -6y - 12 \over 5z \\ -2xy - 4x \over -2y & = & -6y - 12 \over -2y + 5z \\ -x - 4x \over -2y & = & -6y - 12 \over -2y + 5z \\ -x + -4x & = & -6y - 12 \over -2y + 5z \\ -5x & = & -6y - 12 \over -2y + 5z \\ -5x \over -5 & = & -6y - 12 \over -2y + 5z - 5 \\ x & = & -6y - 12 \over -2y + 5z - 5 \\ x & = & 6y + 12 \over 2y - 5z + 5 \\ \end{eqnarray} $$ #solve-for-x-03.solution :escaped $$ \begin{eqnarray} -2xy + 5xz - 5x + 10 & = & -6y - 2 \\ -2xy + 5xz - 5x + 10 - 10 & = & -6y - 2 - 10 \\ -2xy + 5xz - 5x & = & -6y - 12 \\ x \cdot (-2y + 5z - 5) & = & -6y - 12 \\ x \cdot (-2y + 5z - 5) \over -2y + 5z - 5 & = & -6y - 12 \over -2y + 5z - 5 \\ x & = & -6y - 12 \over -2y + 5z - 5 \\ x & = & 6y + 12 \over 2y - 5z + 5 \\ \end{eqnarray} $$ %h2 Solve for $m$: #solve-for-m-01.solution :escaped $$ \begin{eqnarray} -8m - 4n - 8p - 5 & = & -5n - p + 6 \\ -8m - 4n - 8p - 5 + 5 & = & -5n - p + 6 + 5 \\ -8m - 4n - 8p & = & -5n - p + 11 \\ -8m - 4n - 8p + 8p & = & -5n - p + 11 + 8p \\ -8m - 4n & = & -5n + -p + 11 + 8p \\ -8m - 4n & = & -5n + 11 + 7p \\ -8m - 4n + 4n & = & -5n + 11 + 7p + 4n \\ -8m & = & -n + 11 + 7p \\ -8m & = & -n + 11 + 7p \\ -8m \over -8 & = & -n + 11 + 7p \over -8 \\ m & = & -n + 11 + 7p \over -8 \\ m & = & n - 11 - 7p \over 8 \\ m & = & n - 7p - 11 \over 8 \\ \end{eqnarray} $$ %h2 Solve for $x$ (again): #solve-for-x2-01.solution :escaped $$ \begin{eqnarray} 7xy + 4xz + x + 1 & = & y + 6 \\ 7xy + 4xz + x + 1 - 1 & = & y + 6 - 1 \\ 7xy + 4xz + x & = & y + 5 \\ x \cdot (7y + 4z + 1) & = & y + 5 \\ x \cdot (7y + 4z + 1) \over 7y + 4z + 1 & = & y + 5 \over 7y + 4z + 1 \\ x & = & y + 5 \over 7y + 4z + 1 \\ \end{eqnarray} $$ %h2 Solve for $v$: #solve-for-v-01.solution :escaped $$ \begin{eqnarray} 3vw - 2vx + 10v + 1 & = & w + 4 \\ 3vw - 2vx + 10v + 1 - 1 & = & w + 4 - 1 \\ 3vw - 2vx + 10v & = & w + 3 \\ v \cdot (3w - 2x + 10) & = & w + 3 \\ v \cdot (3w - 2x + 10) \over 3w - 2x + 10 & = & w + 3 \over 3w - 2x + 10 \\ v & = & w + 3 \over 3w - 2x + 10 \\ \end{eqnarray} $$