Gaussian Method

1.17 Solving

a.

$$ \begin{flalign}

2x + 3y &= 13 &\\ x - y &= -1\

&\rightarrow (2p2 - p1) &\\

5y &= 15 \ x - y &= -1 \

&\rightarrow (\text{swap } p1 / p2) &\\

x - y &= -1 \ 5y &= 15 \ \ y &= 3 \ x &= 2 \

\end{flalign} $$

b.

$$ \begin{flalign} x - z &= 0 &\\ 3x + y &= 1 \ -x + y + z &= 4\ \ \rightarrow& p3 + p1 \ x - z &= 0 \ 3x + y &= 1 \ y &= 4\ \ x = -1\ y = 4\ z = -1 \end{flalign} $$

Finding type of solutions

1.18

a.

$$ \begin{flalign} −3x + 2y &= 0 &\\ −2y &= 0 \ &\rightarrow \text{One solution} \end{flalign} $$

b.

$$ \begin{flalign} x + y &= 4 &\ y − z &= 0 \ &\rightarrow \text{Infinitely many, now row leading with z} \end{flalign} $$

$$ \begin{flalign} −3x + 2y &= 0 &\\ −2y &= 0 \ &\rightarrow \text{One solution} \end{flalign} $$

1.19

a.

$$ \begin{flalign} 2x + 2y &= 5 &\\ x - 4y &= 0 \

\rightarrow & -\frac{1}{2}p1 + p2 \

2x + 2y &= 5 &\\

• 5y &= -2.5 \ \ y &= \frac{1}{2}\ x &= 2

\end{flalign} $$