User:Olathe
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Systems of equations
You will be given a bunch of equations and asked to find what all the variables equal :
37a + 5b + 11c + 2d = 5e + 35 6a + d + 250e - 9 = 2 240a + 11b + 12c + 4e + 5 = 0 56a + 10b + d + 5e = 2 2c + 30d = 1 - 15e
Get them in the form variables = number.
37a + 5b + 11c + 2d - 5e = 35 6a + d + 250e = 11 240a + 11b + 12c + 4e = -5 56a + 10b + d + 5e = 2 2c + 30d + 15e = 1
Make a table with columns for each of the variables and a column for the number on the other side of each equation :
<math>
\left (
\begin{matrix}
37 & 5 & 11 & 2 & -5 \\
6 & 0 & 0 & 1 & 250 \\
240 & 11 & 12 & 0 & 4 \\
56 & 10 & 0 & 1 & 5 \\
0 & 0 & 2 & 30 & 15
\end{matrix}
\;
\right |
\left.
\begin{matrix}
35 \\
11 \\
-5 \\
2 \\
1
\end{matrix}
\right )
<math>
This is how you want it to look (the stars can be anything) :
<math>
\left (
\begin{matrix}
1 & * & * & * & * \\
0 & 1 & * & * & * \\
0 & 0 & 1 & * & * \\
0 & 0 & 0 & 1 & * \\
0 & 0 & 0 & 0 & 1
\end{matrix}
\;
\right |
\left.
\begin{matrix}
\mbox{*} \\
\mbox{*} \\
\mbox{*} \\
\mbox{*} \\
\mbox{*}
\end{matrix}
\right )
<math>
The important part is that each row starts with a one and it starts a little later than the rows before, even if it looks something like this :
<math>
\left (
\begin{matrix}
1 & * & * & * & * \\
0 & 0 & 0 & 1 & * \\
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0
\end{matrix}
\;
\right |
\left.
\begin{matrix}
\mbox{*} \\
\mbox{*} \\
0 \\
0 \\
0
\end{matrix}
\right )
<math>
Even though the each new row doesn't start exactly one square behind the last one, they all still start with one and start later and later.
You can now use three rules to change this around :
- Change the order of the rows however you want.
- Multiply everything on a row by any number except zero.
- Add or subtract a row from another.