Mohlala oa Potso ea Puisano mabapi le Sisteme ea Litekanyo tse Mola
Mekhoa ea Li-equation tse Kholo (SLE) ke mohopolo oa motheo o rutoang khafetsa lipalo maemong a mahareng le a thuto e phahameng. Ho tseba SLE ho bohlokoa ka lebaka la ts'ebeliso ea eona e pharaletseng mafapheng a fapaneng, ho tloha fisiks le moruo ho ea boenjiniere. Sehloohong sena, re tla tšohla mehlala e 'maloa ea litsamaiso tsa li-equation tse kholo le litharollo tsa tsona. Re tla sebelisa mekhoa ea ho nkela sebaka, ho tlosa le matrix ho nolofatsa kutloisiso.
Lipotso tsa Mehlala le Puisano
Mohlala oa Potso ea 1: Mokhoa oa ho Nkela Sebaka
Potso:
Rarolla tsamaiso e latelang ea li-equation u sebelisa mokhoa oa ho nkela sebaka:
1. \(2x + 3y = 8\)
2. \(x – 2y = -3\)
Tharollo:
1. Mohato oa pele ke ho rarolla e 'ngoe ea li-equation bakeng sa e 'ngoe ea li-variable. Mohlala, re ka rarolla equation ea bobeli bakeng sa \(x\):
\[ x – 2y = -3 \]
\[x = 2y – 3 \]
2. Kenya \(x = 2y – 3\) sebakeng sa equation ya pele:
\[ 2(2y – 3) + 3y = 8 \]
\[ 4y – 6 + 3y = 8 \]
\[ 7y – 6 = 8 \]
\[ 7y = 14 \]
\[ y = 2 \]
3. Jwale, kenya \(y = 2\) ka hara equation \(x = 2y – 3\):
\[x = 2(2) – 3 \]
\[x = 4 – 3 \]
\[x = 1 \]
Kahoo, tharollo ea tsamaiso ea li-equation ke \( x = 1 \) le \( y = 2 \).
Mohlala oa Potso ea 2: Mokhoa oa ho Felisa
Potso:
Rarolla tsamaiso e latelang ea li-equation u sebelisa mokhoa oa ho felisa:
1. \(3x + 2y = 12\)
2. \(5x – y = 9\)
Tharollo:
1. Mohato oa pele ke ho etsa hore coefficient ea e 'ngoe ea li-variable e tšoane ho li-equation ka bobeli. Re ka atisa equation ea bobeli ka 2 ho etsa hore coefficient ea \(y\) e tšoane:
\[ 2(5x – y) = 2(9) \]
\[ 10x – 2y = 18 \]
2. Kenya diekabone tse pedi ho tlosa \(y\):
\[ 3x + 2y + 10x – 2y = 12 + 18 \]
\[ 13x = 30 \]
\[ x = \frac{30}{13} \]
3. Kenya \( x = \frac{30}{13} \) sebakeng sa equation ya pele:
\[ 3\left(\frac{30}{13}\right) + 2y = 12 \]
\[ \frac{90}{13} + 2y = 12 \]
\[ 2y = 12 – \frac{90}{13} \]
\[ 2y = \frac{156}{13} – \frac{90}{13} \]
\[ 2y = \frac{66}{13} \]
\[ y = \frac{33}{13} \]
Kahoo, ditharollo tsa tsamaiso ya di-equation ke \( x = \frac{30}{13} \) le \( y = \frac{33}{13} \).
Mohlala oa Potso ea 3: Mokhoa oa Matrix (Ho Felisoa ha Gaussian)
Potso:
Rarolla sistimi e latelang ea li-equation u sebelisa mokhoa oa matrix:
1. \(x + y + z = 6\)
2. \(2x – y + 3z = 14\)
3. \(4x + 2y – z = 2\)
Tharollo:
1. Sebopeho sa matrix e eketsehileng sa sistimi ea li-equation:
\[ \begin{pmatrix}
1 & 1 & 1 & | & 6 \\
2 le -1 le 3 le | le 14 \\
4 le 2 le -1 le | le 2
\end{pmatrix} \]
2. Ts'ebetso ea ho felisa Gaussian:
– Fetola mola wa bobedi ho sephetho sa mola wa bobedi ka ho tlosa mola wa pele habeli:
\[ \begin{pmatrix}
1 & 1 & 1 & | & 6 \\
0 le -3 le 1 le | le 2 \\
4 le 2 le -1 le | le 2
\end{pmatrix} \]
– Fetola mola wa boraro ho sephetho sa mola wa boraro ntle le makgetlo a mane a mola wa pele:
\[ \begin{pmatrix}
1 & 1 & 1 & | & 6 \\
0 le -3 le 1 le | le 2 \\
0 le -2 le -5 le | le -22
\end{pmatrix} \]
– Fetola mola wa boraro ho ya sephethong sa mola wa boraro hammoho le pedi ho tse tharo tsa mola wa bobedi:
\[ \begin{pmatrix}
1 & 1 & 1 & | & 6 \\
0 le -3 le 1 le | le 2 \\
0 le 0 le -4 le | le -20
\end{pmatrix} \]
– Fetola mola wa boraro ho ya sephethong sa mola wa boraro o arotsweng ka -4:
\[ \begin{pmatrix}
1 & 1 & 1 & | & 6 \\
0 le -3 le 1 le | le 2 \\
0 le 0 le 1 le | le 5
\end{pmatrix} \]
– Fetolela mola wa bobedi ho sephetho sa mola wa bobedi hammoho le mola wa boraro:
\[ \begin{pmatrix}
1 & 1 & 1 & | & 6 \\
0 le -3 le 0 le | le -3 \\
0 le 0 le 1 le | le 5
\end{pmatrix} \]
– Fetola mola wa bobedi ho ya sephethong sa mola wa bobedi o arotsweng ka -3:
\[ \begin{pmatrix}
1 & 1 & 1 & | & 6 \\
0 & 1 & 0 & | & 1 \\
0 le 0 le 1 le | le 5
\end{pmatrix} \]
– Fetola mola wa pele ho sephetho sa mola wa pele ntle le mola wa bobedi le mola wa boraro:
\[ \begin{pmatrix}
1 & 0 & 0 & | & 0 \\
0 & 1 & 0 & | & 1 \\
0 le 0 le 1 le | le 5
\end{pmatrix} \]
Kahoo, ditharollo tsa tsamaiso ya di-equation ke \( x = 0 \), \( y = 1 \), le \( z = 5 \).
Qetello
Ho utloisisa mekhoa ea ho rarolla litsamaiso tsa li-equation tse otlolohileng ho bohlokoa bakeng sa ho tseba lipalo. Mekhoa ea ho nkela sebaka, ho tlosa le matrix e fana ka mekhoa e fapaneng ea ho fumana tharollo e nepahetseng. Ka mokhoa o tsitsitseng le kutloisiso e tiileng ea likhopolo, mang kapa mang a ka tseba mekhoa ena le ho e sebelisa maemong a fapaneng. Re tšepa hore mehlala e tšohloang sehloohong sena e tla thusa babali ho utloisisa hamolemo le ho tseba litsamaiso tsa li-equation tse otlolohileng.
