Muenzaniso weMubvunzo weKukurukurirana paSisitimu yeZviyero Zvakatsetseka
Masisitimu eLinear Equations (SLE) ipfungwa huru inowanzo dzidziswa mumasvomhu padanho rechipiri nerechitatu. Kuziva SLE kwakakosha nekuda kwekushandiswa kwayo munzvimbo dzakasiyana-siyana, kubva pafizikisi nehupfumi kusvika kumainjiniya. Muchinyorwa chino, tichakurukura mienzaniso yakati wandei yemasisitimu eLinear equations nemhinduro dzawo. Tichashandisa nzira dzekutsiva, kubvisa, uye matrix kuti zvive nyore kunzwisisa.
Mibvunzo yemuenzaniso nekukurukurirana
Muenzaniso Mubvunzo 1: Nzira yekuchinjana
Mubvunzo:
Gadzirisa nzira inotevera yekuenzanisa uchishandisa nzira yekutsiva:
1. \(2x + 3y = 8\)
2. \(x – 2y = -3\)
Mhinduro:
1. Danho rekutanga nderekugadzirisa imwe ye equation yeimwe ye variables. Semuenzaniso, tinogona kugadzirisa equation yechipiri ye \(x\):
\[ x – 2y = -3 \]
\[x = 2y – 3 \]
2. Isa \(x = 2y – 3\) muequation yekutanga:
\[ 2(2y – 3) + 3y = 8 \]
\[ 4y – 6 + 3y = 8 \]
\[makore 7 – 6 = 8 \]
\[7y = 14 \]
\[ y = 2 \]
3. Zvino, chinja \(y = 2\) muequation \(x = 2y – 3\):
\[x = 2(2) – 3 \]
\[x = 4 – 3 \]
\[x = 1 \]
Saka, mhinduro yehurongwa hwema equation ndeye \( x = 1 \) uye \( y = 2 \).
Muenzaniso Mubvunzo 2: Nzira Yekubvisa
Mubvunzo:
Gadzirisa nzira inotevera yekuenzanisa uchishandisa nzira yekubvisa:
1. \(3x + 2y = 12\)
2. \(5x – y = 9\)
Mhinduro:
1. Danho rekutanga nderekuita kuti coefficient yeimwe yevariables ive yakafanana mu equation dzese dziri mbiri. Tinogona kuwanza equation yechipiri ne2 kuti coefficient ye \(y\) ive yakafanana:
\[ 2(5x – y) = 2(9) \]
\[ 10x – 2y = 18 \]
2. Wedzera ma equation maviri kubvisa \(y\):
\[ 3x + 2y + 10x – 2y = 12 + 18 \]
\[ 13x = 30 \]
\[ x = \frac{30}{13} \]
3. Isa \( x = \frac{30}{13} \) muequation yekutanga:
\[ 3\kuruboshwe(\frac{30}{13}\kurudyi) + 2y = 12 \]
\[ \frac{90}{13} + 2y = 12 \]
\[ 2y = 12 – \frac{90}{13} \]
\[ 2y = \frac{156}{13} – \frac{90}{13} \]
\[ 2y = \frac{66}{13} \]
\[ y = \frac{33}{13} \]
Saka, mhinduro dzehurongwa hwema equation ndi \( x = \frac{30}{13} \) uye \( y = \frac{33}{13} \).
Muenzaniso Mubvunzo 3: Nzira yeMatrix (Gaussian Elimination)
Mubvunzo:
Gadzirisa hurongwa hunotevera hwema equation uchishandisa nzira yematrix:
1. \(x + y + z = 6\)
2. \(2x – y + 3z = 14\)
3. \(4x + 2y – z = 2\)
Mhinduro:
1. Chimiro che matrix chakawedzerwa chehurongwa hwe equations:
\[ \begin{pmatrix}
1 & 1 & 1 & | & 6 \\
2 & -1 & 3 & | & 14 \\
4 & 2 & -1 & | & 2
\kuguma{pmatrix} \]
2. Maitiro ekubvisa Gaussian:
– Chinja mutsara wechipiri kuita mhedzisiro yemutsara wechipiri kubvisa kaviri mutsara wekutanga:
\[ \begin{pmatrix}
1 & 1 & 1 & | & 6 \\
0 & -3 & 1 & | & 2 \\
4 & 2 & -1 & | & 2
\kuguma{pmatrix} \]
– Chinja mutsara wechitatu kuita mhedzisiro yemutsara wechitatu kubvisa kana pane wekutanga:
\[ \begin{pmatrix}
1 & 1 & 1 & | & 6 \\
0 & -3 & 1 & | & 2 \\
0 & -2 & -5 & | & -22
\kuguma{pmatrix} \]
– Chinja mutsara wechitatu kuita mhedzisiro yemutsara wechitatu pamwe chete nezvikamu zviviri kubva muzvitatu zvemutsara wechipiri:
\[ \begin{pmatrix}
1 & 1 & 1 & | & 6 \\
0 & -3 & 1 & | & 2 \\
0 & 0 & -4 & | & -20
\kuguma{pmatrix} \]
– Chinja mutsara wechitatu kuita mhedzisiro yemutsara wechitatu wakakamurwa ne -4:
\[ \begin{pmatrix}
1 & 1 & 1 & | & 6 \\
0 & -3 & 1 & | & 2 \\
0 & 0 & 1 & | & 5
\kuguma{pmatrix} \]
- Chinja mutsara wechipiri kuita mhedzisiro yemutsara wechipiri pamwe nemutsara wechitatu:
\[ \begin{pmatrix}
1 & 1 & 1 & | & 6 \\
0 & -3 & 0 & | & -3 \\
0 & 0 & 1 & | & 5
\kuguma{pmatrix} \]
– Chinja mutsara wechipiri kuita mhedzisiro yemutsara wechipiri wakakamurwa ne -3:
\[ \begin{pmatrix}
1 & 1 & 1 & | & 6 \\
0 & 1 & 0 & | & 1 \\
0 & 0 & 1 & | & 5
\kuguma{pmatrix} \]
- Chinja mutsara wekutanga kuita mhedzisiro yemutsara wekutanga uchibvisa mutsara wechipiri nemutsara wechitatu:
\[ \begin{pmatrix}
1 & 0 & 0 & | & 0 \\
0 & 1 & 0 & | & 1 \\
0 & 0 & 1 & | & 5
\kuguma{pmatrix} \]
Saka, mhinduro dzehurongwa hwema equation ndi \( x = 0 \), \( y = 1 \), uye \( z = 5 \).
Mhedziso
Kunzwisisa nzira dzekugadzirisa masisitimu e-linear equation kwakakosha pakudzidza masvomhu. Nzira dzekutsiva, kubvisa, uye matrix dzinopa nzira dzakasiyana dzekuwana mhinduro chaiyo. Nekudzidzira kwakasimba uye kunzwisisa kwakasimba kwepfungwa, chero munhu anogona kugona matekiniki aya uye kuashandisa mumamiriro akasiyana-siyana. Tinovimba kuti mienzaniso iri kukurukurwa munyaya ino ichabatsira vaverengi kunzwisisa zviri nani uye kugona masisitimu e-linear equation.
