1.1 KiB
1.1 KiB
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}