Khopolo ea Litekanyo tse Moqotetsane
Li-equation tse otlolohileng ke mohopolo oa motheo lipalo tse nang le lits'ebetso tse ngata saenseng, boenjiniere, moruo le masimo a mang a mangata. Ho utloisisa li-equation tse otlolohileng ke senotlolo sa ho rarolla mathata a mangata a lefats'e la nnete a kenyeletsang likamano tse otlolohileng lipakeng tsa mefuta-futa. Sengoloa sena se tla hlalosa mohopolo oa li-equation tse otlolohileng, mokhoa oa ho li rarolla, le tse ling tsa lits'ebetso tsa tsona tse sebetsang.
Tlhaloso ea Litekanyo tse Moqotetsane
Equation e otlolohileng ke equation e kenyeletsang diphetoho tse le nngwe kapa tse ngata tse nang le matla a hodimo a phetoho e le nngwe. Sebopeho se akaretsang sa equation e otlolohileng e nang le phetoho e le nngwe se ka ngolwa tjena:
\[ selepe + b = 0 \]
moo \( a \) le \( b \) e leng di-constant, mme \( x \) ke phetoho.
Bakeng sa equation e otlolohileng e nang le di-variable tse pedi, sebopeho se akaretsang ke:
\[ selepe + ka + c = 0 \]
moo \( a \), \( b \), le \( c \) e leng di-constant, mme \( x \) le \( y \) ke di-variable.
Moelelong o akaretsang haholoanyane, di-equation tse otlolohileng di ka kenyeletsa di-variable tse fetang tse pedi mme di ka ngolwa ka sebopeho sa matrix.
Mehlala ea Litekanyo tse Lekanang ho Fetoha ho le Leng
Nahana ka equation:
\[ 3x – 5 = 0 \]
Ho rarolla sena, re hloka ho fumana boleng ba \( x \) bo etsang hore equation e be nnete. Tabeng ena, re fetisetsa constant lehlakoreng le letona la equation:
\[ 3x = 5 \]
Ebe, arola mahlakore ka bobedi ka coefficient ya \( x \):
\[ x = \frac{5}{3} \]
Kahoo, tharollo ea equation \( 3x – 5 = 0 \) ke \( x = \frac{5}{3} \).
Mehlala ea Litekanyo tse Moeli ka Mefuta-futa e 'Meli
Nahana ka equation:
\[ 2x + 3y – 6 = 0 \]
Tekanyo ena e hlalosa mola o ka sefofaneng sa Cartesian sa mahlakore a mabedi. Ho hlalosa mola ona, re ka fumana dintlha tsa ona tsa kopano le mothapo wa x le mothapo wa y.
Bakeng sa x-intercept (moo \( y = 0 \)):
\[ 2x – 6 = 0 \]
\[ 2x = 6 \]
\[x = 3 \]
Bakeng sa y-intercept (moo \( x = 0 \)):
\[ 3y – 6 = 0 \]
\[ 3y = 6 \]
\[ y = 2 \]
Kahoo, mola ona o feta lintlheng (3, 0) le (0, 2).
Ho rarolla Sistimi ea Litekanyo tse Kholo
Hangata, re tobana le litsamaiso tsa li-equation tse otlolohileng, e leng sete ea li-equation tse otlolohileng tse lokelang ho rarolloa ka nako e le 'ngoe. Ho na le mekhoa e 'maloa e ka sebelisoang ho rarolla litsamaiso tsa li-equation tse otlolohileng, ho kenyeletsoa:
1. Mokhoa oa ho Nkela Sebaka
Mokhoa oa ho nkela sebaka o kenyelletsa ho rarolla e 'ngoe ea li-equation bakeng sa phetoho e le 'ngoe, ebe u kenya sephetho ho equation e 'ngoe. Mohlala, nahana ka tsamaiso e latelang ea li-equation:
\[ 2x + y = 5 \]
\[ x – 2y = -4 \]
Taba ea pele, re rarolla equation ea pele bakeng sa \( y \):
\[ y = 5 – 2x \]
Ebe re kenya \( y \) sebakeng sa equation ea bobeli:
\[ x – 2(5 – 2x) = -4 \]
\[ x – 10 + 4x = -4 \]
\[ 5x – 10 = -4 \]
\[ 5x = 6 \]
\[ x = \frac{6}{5} \]
Ebe re nkela boleng ba \( x \) sebaka ka har'a equation \( y = 5 - 2x \):
\[ y = 5 – 2\left( \frac{6}{5} \right) \]
\[ y = 5 – \frac{12}{5} \]
\[ y = \frac{25}{5} – \frac{12}{5} \]
\[ y = \frac{13}{5} \]
Kahoo, tharollo ea tsamaiso ea li-equation ke \( x = \frac{6}{5} \) le \( y = \frac{13}{5} \).
2. Mokhoa oa ho Felisa
Mokhoa oa ho tlosa o kenyelletsa ho eketsa kapa ho tlosa li-equation ho tlosa e 'ngoe ea li-variable. Nahana ka sistimi ea li-equation:
\[ 3x + 2y = 8 \]
\[ 2x – 3y = -1 \]
Ho tlosa \( y \), re ka eketsa di-equation ka mora ho atisa e 'ngoe le e 'ngoe ka coefficient e loketseng:
Atisa equation ea pele ka 3 le equation ea bobeli ka 2:
\[ 9x + 6y = 24 \]
\[ 4x – 6y = -2 \]
Ebe o eketsa di-equation tse pedi:
\[ 13x = 22 \]
\[ x = \frac{22}{13} \]
Kenya boleng ba \( x \) sebakeng sa e 'ngoe ea li-equation tsa pele ho fumana \( y \):
\[ 3\left( \frac{22}{13} \right) + 2y = 8 \]
\[ \frac{66}{13} + 2y = 8 \]
\[ 2y = 8 – \frac{66}{13} \]
\[ 2y = \frac{104}{13} – \frac{66}{13} \]
\[ 2y = \frac{38}{13} \]
\[ y = \frac{19}{13} \]
Kahoo, tharollo ea tsamaiso ea li-equation ke \( x = \frac{22}{13} \) le \( y = \frac{19}{13} \).
3. Mokhoa oa Matrix (Ho Felisoa ha Gaussian)
Mokhoeng ona, re sebelisa matrices ho laola tsamaiso ea li-equation e le hore li ka rarolloa ka mokhoa o hlophisehileng haholoanyane. Mohlala, ho rarolla tsamaiso:
\[ 3x + 2y = 8 \]
\[ 2x – 3y = -1 \]
Re ka e ngola ka mokhoa oa matrix e eketsehileng:
\[ \begin{pmatrix}
3 le 2 le | le 8\\
2 le -3 le | le -1
\end{pmatrix} \]
Mohato o latelang ke ho sebedisa mesebetsi ya motheo ya mela ho rarolla tsamaiso ena. Leha ho le jwalo, ka lebaka la ho rarahana ha dintlha tsa mokgwa ona, ho hlokahala thuto e tebileng haholoanyane bakeng sa kutlwisiso e felletseng.
4. Mokhoa oa Litšoantšo
Mokhoa oa litšoantšo o re lumella ho fumana litharollo ka ho rala equation sefofaneng sa coordinate le ho fumana lintlha tsa moo li-graph li kopanang teng. Mohlala, bakeng sa sistimi:
\[ y = 2x + 1 \]
\[ y = -x + 3 \]
Re taka mela ena e 'meli holim'a sefofane sa xy 'me re fumana ntlha eo mela e 'meli e kopanang ho eona, e leng tharollo ea tsamaiso ea li-equation.
Litšebeliso tsa Litekanyo tse Kholo
Li-equation tse otlolohileng le litsamaiso tsa li-equation tse otlolohileng li na le lits'ebetso tse pharaletseng masimong a fapaneng, a mang a ona a kenyeletsa:
1. Moruo
Moruong, di-equation tse otlolohileng di sebediswa ho sekaseka tekano pakeng tsa phepelo le tlhoko, ho fumana ditheko le bongata ba tekano, le ho etsa mohlala wa diketsahalo tse fapaneng tsa moruo.
2. Boenjiniere le Fisiks
Boenjiniereng, di-equation tse otlolohileng di sebediswa tlhahlobong ya potoloho ya motlakase, tlhahlobo ya sebopeho le thepa, le ditshebediso tse ding tse fapaneng tse kenyeletsang dikamano tse lekanang pakeng tsa di-variable tsa mmele.
3. Mahlale a Sechaba
Li-equation tse otlolohileng hangata li sebelisoa mahlaleng a kahisano ho leka likamano lipakeng tsa lintho tse fetohang, joalo ka tlhahlobo ea regression lipalo-palong.
4. Saense ea Khomphutha
Hangata li-algorithms tsa ntlafatso li kenyelletsa ho rarolla litsamaiso tsa li-equation tse otlolohileng, mohlala, tlhahlobong ea data, ho ithuta ha mochini le lipatlisisong tsa ts'ebetso.
Qetello
Li-equation tse otlolohileng ke mohopolo oa motheo oa lipalo o nang le lits'ebetso tse pharaletseng. Ho utloisisa mokhoa oa ho rarolla li-equation tse otlolohileng le litsamaiso tsa li-equation tse otlolohileng ho bohlokoa bakeng sa masimo a tlohang moruong le boenjiniere ho isa ho mahlale a sechaba. Re sebelisa lisebelisoa tse kang ho nkela sebaka, ho tlosa, le ts'ebeliso ea matrices, re ka rarolla mathata a fapaneng a amanang le likamano tse otlolohileng lipakeng tsa lintho tse fetohang. Ho tseba li-equation tse otlolohileng ho boetse ho bula monyako oa kutloisiso e tebileng ea lipalo le lits'ebetso tsa tsona tsa lefats'e la 'nete.
