Chitsanzo cha Funso Lokambirana pa Dongosolo la Ma Equation Olunjika
Machitidwe a Ma equation a Linear (SLE) ndi mfundo yofunikira yomwe imaphunzitsidwa kawirikawiri mu masamu pamlingo wa sekondale ndi wachitatu. Kudziwa bwino SLE n'kofunika kwambiri chifukwa imagwiritsidwa ntchito kwambiri m'magawo osiyanasiyana, kuyambira pa fizikisi ndi zachuma mpaka uinjiniya. M'nkhaniyi, tikambirana zitsanzo zingapo za machitidwe a ma equation a linear ndi mayankho awo. Tidzagwiritsa ntchito njira zosinthira, kuchotsa, ndi matrix kuti tithandize kumvetsetsa.
Mafunso ndi Kukambirana Zitsanzo
Chitsanzo Funso 1: Njira Yosinthira
Funso:
Konzani njira yotsatirayi ya ma equation pogwiritsa ntchito njira yosinthira:
1. \(2x + 3y = 8\)
2. \(x – 2y = -3\)
Yankho:
1. Gawo loyamba ndikuthetsa imodzi mwa ma equation a chimodzi mwa ma variable. Mwachitsanzo, tikhoza kuthetsa equation yachiwiri ya \(x\):
\[ x – 2y = -3 \]
\[x = 2y – 3 \]
2. Lowetsani \(x = 2y – 3\) mu equation yoyamba:
\[ 2(2y – 3) + 3y = 8 \]
\[ 4y – 6 + 3y = 8 \]
\[ 7y – 6 = 8 \]
\[ 7y = 14 \]
\[ y = 2 \]
3. Tsopano, sinthani \(y = 2\) mu equation \(x = 2y – 3\):
\[x = 2(2) – 3 \]
\[x = 4 – 3 \]
\[x = 1 \]
Kotero, yankho la dongosolo la ma equation ndi \( x = 1 \) ndi \( y = 2 \).
Chitsanzo Funso 2: Njira Yochotsera
Funso:
Konzani njira yotsatirayi ya ma equation pogwiritsa ntchito njira yochotsera:
1. \(3x + 2y = 12\)
2. \(5x – y = 9\)
Yankho:
1. Gawo loyamba ndi kupanga coefficient ya chimodzi mwa zinthu zomwe zili mu ma equation onse awiri kukhala yofanana. Tikhoza kuchulukitsa equation yachiwiri ndi 2 kuti coefficient ya \(y\) ikhale yofanana:
\[ 2(5x – y) = 2(9) \]
\[ 10x – 2y = 18 \]
2. Onjezani ma equation awiriwa kuti muchotse \(y\):
\[ 3x + 2y + 10x – 2y = 12 + 18 \]
\[ 13x = 30 \]
\[ x = \frac{30}{13} \]
3. Lowetsani \( x = \frac{30}{13} \) mu equation yoyamba:
\[ 3\left(\frac{30}{13}\right) + 2y = 12 \]
\[ \frac{90}{13} + 2y = 12 \]
\[ 2y = 12 – \frac{90}{13} \]
\[ 2y = \frac{156}{13} – \frac{90}{13} \]
\[ 2y = \frac{66}{13} \]
\[ y = \frac{33}{13} \]
Kotero, mayankho a dongosolo la ma equation ndi \( x = \frac{30}{13} \) ndi \( y = \frac{33}{13} \).
Chitsanzo Funso 3: Njira ya Matrix (Kuchotsa Gaussian)
Funso:
Konzani njira yotsatirayi ya ma equation pogwiritsa ntchito njira ya matrix:
1. \(x + y + z = 6\)
2. \(2x – y + 3z = 14\)
3. \(4x + 2y – z = 2\)
Yankho:
1. Fomu ya matrix yowonjezereka ya dongosolo la ma equation:
\[ \begin{pmatrix}
1 & 1 & 1 & | & 6 \\
2 & -1 & 3 & | & 14 \\
4 & 2 & -1 & | & 2
\end{pmatrix} \]
2. Njira yochotsera Gaussian:
– Sinthani mzere wachiwiri kukhala zotsatira za mzere wachiwiri kuchotsa kawiri mzere woyamba:
\[ \begin{pmatrix}
1 & 1 & 1 & | & 6 \\
0 & -3 & 1 & | & 2 \\
4 & 2 & -1 & | & 2
\end{pmatrix} \]
– Sinthani mzere wachitatu kukhala zotsatira za mzere wachitatu kupatulapo kanayi kuposa mzere woyamba:
\[ \begin{pmatrix}
1 & 1 & 1 & | & 6 \\
0 & -3 & 1 & | & 2 \\
0 & -2 & -5 & | & -22
\end{pmatrix} \]
– Sinthani mzere wachitatu kukhala zotsatira za mzere wachitatu kuphatikiza magawo awiri mwa atatu a mzere wachiwiri:
\[ \begin{pmatrix}
1 & 1 & 1 & | & 6 \\
0 & -3 & 1 & | & 2 \\
0 & 0 & -4 & | & -20
\end{pmatrix} \]
– Sinthani mzere wachitatu kukhala zotsatira za mzere wachitatu wogawidwa ndi -4:
\[ \begin{pmatrix}
1 & 1 & 1 & | & 6 \\
0 & -3 & 1 & | & 2 \\
0 & 0 & 1 & | & 5
\end{pmatrix} \]
– Sinthani mzere wachiwiri kukhala zotsatira za mzere wachiwiri kuphatikiza mzere wachitatu:
\[ \begin{pmatrix}
1 & 1 & 1 & | & 6 \\
0 & -3 & 0 & | & -3 \\
0 & 0 & 1 & | & 5
\end{pmatrix} \]
– Sinthani mzere wachiwiri kukhala zotsatira za mzere wachiwiri wogawidwa ndi -3:
\[ \begin{pmatrix}
1 & 1 & 1 & | & 6 \\
0 & 1 & 0 & | & 1 \\
0 & 0 & 1 & | & 5
\end{pmatrix} \]
– Sinthani mzere woyamba kukhala zotsatira za mzere woyamba kupatula mzere wachiwiri ndi mzere wachitatu:
\[ \begin{pmatrix}
1 & 0 & 0 & | & 0 \\
0 & 1 & 0 & | & 1 \\
0 & 0 & 1 & | & 5
\end{pmatrix} \]
Kotero, mayankho a dongosolo la ma equation ndi \( x = 0 \), \( y = 1 \), ndi \( z = 5 \).
Mapeto
Kumvetsetsa njira zothetsera machitidwe a ma equation olunjika ndikofunikira kwambiri pakumvetsetsa masamu. Njira zosinthira, kuchotsa, ndi matrix zimapereka njira zosiyanasiyana zopezera yankho lolondola. Ndi machitidwe okhazikika komanso kumvetsetsa bwino malingaliro, aliyense akhoza kudziwa njira izi ndikuzigwiritsa ntchito m'malo osiyanasiyana. Tikukhulupirira kuti zitsanzo zomwe zafotokozedwa m'nkhaniyi zithandiza owerenga kumvetsetsa bwino ndikudziwa machitidwe a ma equation olunjika.
