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}
$$