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I am reading a book called "Linear algebra and its applications" by Gilbert Strang.

Chapter 1 is called “Matrices and Gaussian elimination”, the example in 1.1 I understand, Strang goes through the steps used.

Can someone please explain the steps to solving the two linear equations below using Gaussian elimination?

It's quite obvious that $x = 2$ and $y = 3$ but I'm struggling the understand the steps using Gaussian elimination to work it out.

begincases
2x - y = 5 \[4px]
x + y = 1
endcases

There is something wrong with the given solution, indeed we have that

• $2x - y = 5$


• $x + y = 1 iff 2x+2y=2$

then subtracting the first one from the second we obtain the equivalent system

• $2x - y = 5$


• $0+3y=-3 implies y=-1$

and then

• $x=2$


• $y=-1$

It depends on what kind of Gaussian elimination you use.

If you use Gauss-Crout with also backward elimination, the steps are
beginalign
left[beginarraycc
2 & -1 & 5 \
1 & 1 & 1
endarrayright]
&to
left[beginarraycc
1 & -1/2 & 5/2 \
1 & 1 & 1
endarrayright]
&& R_1getstfrac12R_1
\[6px] &to
left[beginarraycc
1 & -1/2 & 5/2 \
0 & 3/2 & -3/2
endarrayright]
&& R_2gets R_2-R_1
\[6px] &to
left[beginarraycc
1 & -1/2 & 5/2 \
0 & 1 & -1
endarrayright]
&& R_2gets tfrac23R_2
\[6px] &to
left[beginarraycc
1 & 0 & 2 \
0 & 1 & -1
endarrayright]
&& R_1gets R_1+tfrac12R_2
endalign
Therefore $x=2$ and $y=-1$.