MATHHX B
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#2,,\LWRsiunitxENDTWO }}\)
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9.6 Beviser til lineær programmering
-
Sætning 9.3.1
Lad \(f(x,y)=ax+by+c\), antag at \(b\neq 0\) og lad \(t\) være et tal. Da er niveaukurven \(N(t)\) en linje med forskriften:
\[y=-\frac {a}{b}x+\frac {t-c}{b}\]
Ændres \(t\) vil linjen parallelforskydes i lodret regning. Der gælder:
-
Bevis
Niveaukurven \(N(t)\) er givet ved ligningen
\[f(x,y)=t\]
Vi indsætter forskriften for \(f\)
\[ax+by+c=t\]
Vi trækker \(ax\) og \(c\) fra på begge sider
\[by=t-ax-c\]
Vi omskriver højresiden
\[by=-ax + t-c\]
Nu divideres med \(b\) på begge sider (og det går godt fordi vi har antaget at \(b\neq 0\)) .
\[y= \frac {-ax +t-c}{b}\]
Vi deler brøken op:
\[y=\frac {-ax}{b}+\frac {t-c}{b}\]
Vi omskriver første brøk:
\[y=-\frac {a}{b}x+\frac {t-c}{b}\]
Vi ser at niveaukurven har form som en lineær funktion med hældningen \(-\frac {a}{b}\). Hældningen ikke af \(t\), og niveaulinjerne må derfor være parallelle.
Vi ser, at niveaulinjerne skærer \(y\)-aksen i \(\frac {t-c}{b}\). Her vil tælleren blive større, hvis \(t\) bliver større. Hvis \(b>0\) må det betyde, at hele brøken bliver større og linjen vil dermed blive forskudt op ad. Hvis
\(b<0\) vil brøken blive mindre når tælleren er bliver større, og derfor vil linjen blive forskudt nedad, når \(t\) bliver større.