Ho Rarolla Litekanyo tse Tšoanang
Li-equation tsa nako e le 'ngoe, tse tsejoang hape e le litsamaiso tsa li-equation, ke lihlopha tsa li-equation tse nang le li-variable tse ngata. Li-equation tsena li bitsoa nako e le 'ngoe hobane li rarolloa hammoho, ho bolelang hore tharollo e tlameha ho khotsofatsa li-equation tsohle ka nako e le 'ngoe. Ho rarolla li-equation tsa nako e le 'ngoe ke bokhoni ba motheo ba lipalo bo fumanang lits'ebetso mafapheng a fapaneng a kang fisiks, moruo, boenjiniere le saense ea khomphutha. Sengoloa sena se ikemiselitse ho fana ka kakaretso e felletseng ea mekhoa e sebelisoang hangata ho rarolla li-equation tsa nako e le 'ngoe.
Mehopolo ea Motheo
Pele o teba mekhoeng ea ho rarolla li-equation ka nako e le 'ngoe, ho bohlokoa ho utloisisa mantsoe le likhopolo tse ling tsa motheo.
– Diphetoho: Matshwao a emelang boleng bo sa tsejweng, hangata a bontshwang ka ditlhaku tse kang x, y, le z.
– Litekanyo tse Mola: Litekanyo tseo ho tsona diphetoho di phahamiswang ho ya matleng a e nngwe mme di hlaha ka sebopeho se mola (mohlala, \(2x + 3y = 6\)).
– Ditekanyo tse seng tsa mola: Ditekanyo tse kenyeletsang diphetoho tse phahamiseditsweng matleng a mang ntle le a le mong, tse kenyeletsang dihlahiswa tsa diphetoho, mesebetsi ya trigonometric, jj. (mohlala, \(x^2 + y^2 = 9\)).
– Mekhoa ea Li-equation: Lihlopha tsa li-equation tse peli kapa ho feta tse arolelanang sete e tšoanang ea li-variable.
A re nahaneng ka mohlala o tloaelehileng oa sistimi ea li-equation tse otlolohileng:
\[
\ qala{maemo}
2x + 3y = 6 \\
4x – y = 5
\qetela{maemo}
\]
Tharollo ea sistimi ena ke para efe kapa efe \((x, y)\) e khotsofatsang li-equation ka bobeli ka nako e le 'ngoe.
Mekhoa ea ho Rarolla Litekanyo tse Tšoanang
1. Mokhoa oa Litšoantšo
Mokhoa oa litšoantšo o kenyelletsa ho rala equation ka 'ngoe holim'a gridi ea coordinate le ho khetholla lintlha tseo li-graph li kopanang ho tsona. Lintlha tsa li-contraction li emela litharollo tsa sistimi ea li-equation.
mehato:
1. Fetolela equation e 'ngoe le e 'ngoe ho ea ho foromo ea \( y = mx + c \) moo \( m \) e leng leralla 'me \( c \) e le y-intercept.
2. Rala mela e emeloang ke di-equation tsena kerafong.
3. Fumana moo mela e kopanang teng.
mohlala:
Nahana ka tsamaiso ea li-equation:
\[
\ qala{maemo}
2x + 3y = 6 \\
4x – y = 5
\qetela{maemo}
\]
Fetola tsena hore e be sebopeho sa ho thibela leralla:
\[
\ qala{maemo}
y = -\frac{2}{3}x + 2 \\
y = 4x - 5
\qetela{maemo}
\]
Rala mela kerafong ho fumana ntlha ya kgokelo e emelang tharollo. Mohlaleng ona, tharollo ke \((x, y) = (1.5, 1)\).
2. Mokhoa oa ho Phetela
Mokhoa oa ho nkela sebaka o kenyelletsa ho rarolla e 'ngoe ea li-equation bakeng sa phetoho e le 'ngoe le ho kenya polelo eo sebakeng sa equation e 'ngoe.
mehato:
1. Rarolla e 'ngoe ea li-equation bakeng sa phetoho e le 'ngoe.
2. Kenya polelo ena sebakeng sa equation e 'ngoe, e leng se fellang ka equation e le 'ngoe e nang le phetoho e le 'ngoe.
3. Rarolla equation ena e feto-fetohang e le 'ngoe.
4. Kenya boleng bo fumanoeng sebakeng sa polelo e fumanoang mohatong oa 1 ho fumana boleng ba phetoho ea bobeli.
mohlala:
Nahana ka sistimi:
\[
\ qala{maemo}
2x + 3y = 6 \\
4x – y = 5
\qetela{maemo}
\]
Rarolla equation ea pele bakeng sa \( y \):
\[
y = 2 – \frac{2}{3}x
\]
Kenya polelo ena sebakeng sa equation ea bobeli:
\[
4x – (2 – \frac{2}{3}x) = 5
\]
Nolofatsa le ho rarolla bakeng sa \( x \):
\[
4x – 2 + \frac{2}{3}x = 5 \\
\frac{14x}{3} = 7 \\
x = \frac{3}{2}
\]
Kenya sebaka sa \( x = \frac{3}{2} \) ho \( y = 2 – \frac{2}{3}x \):
\[
y = 2 – \frac{2}{3} \times \frac{3}{2} = 2 – 1 = 1
\]
Kahoo, tharollo ke \( (x, y) = \left(\frac{3}{2}, 1\right) \).
3. Mokhoa oa ho Felisa
Mokhoa oa ho felisa o kenyelletsa ho eketsa kapa ho tlosa li-equation ho felisa phetoho e le 'ngoe, e leng se etsang hore ho khonehe ho rarolla phetoho e setseng.
mehato:
1. Atisa equation e le 'ngoe kapa ka bobeli ka e sa fetoheng e le hore li-coefficient tsa e 'ngoe ea li-variable li fapane.
2. Eketsa kapa tlosa di-equation ho tlosa phetoho e le nngwe.
3. Rarolla equation e hlahang bakeng sa di-variable tse setseng.
4. Kenya boleng bona sebakeng sa e 'ngoe ea li-equation tsa pele ho fumana boleng ba phetoho e tlositsoeng.
mohlala:
Nahana ka sistimi:
\[
\ qala{maemo}
2x + 3y = 6 \\
4x – y = 5
\qetela{maemo}
\]
Atisa equation ea bobeli ka 3:
\[
4x – y = 5 \\
12x - 3y = 15
\]
Kenya equation ea bobeli e fetotsoeng ho equation ea pele:
\[
2x + 3y + 12x – 3y = 6 + 15 \\
14x = 21 \\
x = \frac{21}{14} = \frac{3}{2}
\]
Kenya \( x = \frac{3}{2} \) sebakeng sa e 'ngoe ea li-equation tsa pele:
\[
2 \left(\frac{3}{2}\right) + 3y = 6 \\
3 + 3y = 6 \\
3y = 3 \\
y =1
\]
Kahoo, tharollo ke \( \left(x, y\right) = \left(\frac{3}{2}, 1\right) \).
4. Mokhoa oa Matrix (Ho Felisoa ha Gaussian le ho Fetoha)
Bakeng sa litsamaiso tse kholoanyane, mekhoa ea matrix joalo ka ho tlosa Gaussian kapa ho sebelisa matrix inverse e ka ba molemo haholoanyane.
Ho Felisoa ha Gaussian:
1. Ngola sistimi e le matrix e ekelitsoeng.
2. Sebelisa mesebetsi ea mola ho fetolela matrix ho ba sebopeho sa echelon ea mola.
3. E nke sebaka sa ho fumana tharollo.
mohlala:
Sistimi:
\[
\ qala{maemo}
2x + 3y = 6 \\
4x – y = 5
\qetela{maemo}
\]
Matrix e eketsehileng:
\[
\begin{pmatrix}
2 le 3 le | le 6 \\
4 le -1 le | le 5
\end{pmatrix}
\]
Ts'ebetso ea mola ho ea ho sebopeho sa echelon ea mola:
1. \( R2 \motsu o leqeleng R2 – 2R1 \):
\[
\begin{pmatrix}
2 le 3 le | le 6 \\
0 le -7 le | le -7
\end{pmatrix}
\]
Moemeli oa morao:
\[
-7y = -7 \e bolela y = 1 \\
2x + 3(1) = 6 \e fana ka maikutlo a 2x =
