95 lines
1.1 KiB
Markdown
95 lines
1.1 KiB
Markdown
# Gaussian Method
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## 1.17 Solving
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### a.
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$$
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\begin{flalign}
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2x + 3y &= 13 &\\\
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x - y &= -1\\
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&\rightarrow (2p2 - p1) &\\\
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5y &= 15 \\
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x - y &= -1 \\
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&\rightarrow (\text{swap } p1 / p2) &\\\
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x - y &= -1 \\
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5y &= 15 \\
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\\
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y &= 3 \\
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x &= 2 \\
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\end{flalign}
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$$
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### b.
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$$
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\begin{flalign}
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x - z &= 0 &\\\
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3x + y &= 1 \\
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-x + y + z &= 4\\
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\\
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\rightarrow& p3 + p1 \\
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x - z &= 0 \\
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3x + y &= 1 \\
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y &= 4\\
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\\
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x = -1\\
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y = 4\\
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z = -1
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\end{flalign}
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$$
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## Finding type of solutions
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## 1.18
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### a.
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$$
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\begin{flalign}
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−3x + 2y &= 0 &\\\
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−2y &= 0 \\
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&\rightarrow \text{One solution}
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\end{flalign}
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$$
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### b.
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$$
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\begin{flalign}
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x + y &= 4 &\\
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y − z &= 0 \\
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&\rightarrow \text{Infinitely many, now row leading with z}
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\end{flalign}
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$$
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$$
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\begin{flalign}
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−3x + 2y &= 0 &\\\
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−2y &= 0 \\
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&\rightarrow \text{One solution}
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\end{flalign}
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$$
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## 1.19
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## a.
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$$
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\begin{flalign}
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2x + 2y &= 5 &\\\
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x - 4y &= 0 \\
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\rightarrow & -\frac{1}{2}p1 + p2 \\
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2x + 2y &= 5 &\\\
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- 5y &= -2.5 \\
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\\
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y &= \frac{1}{2}\\
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x &= 2
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\end{flalign}
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$$
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