Whakaoti i te {l}{20-7600x-1600y=0}{20-1000x-6000y=0}

$$\left\{ \begin{array} { l } { 20 - 7600 x - 1600 y = 0 } \\ { 20 - 1000 x - 6000 y = 0 } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{1}{500}=0.002$$

$$y=\frac{3}{1000}=0.003$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Hei whakaoti i ētahi whārite takirua mā te whakakapinga, me whakaoti tētahi whārite i te tuatahi mō tētahi o ngā taurangi. Ka whakakapi i te otinga mō taua taurangi ki tērā o ngā whārite.

$$-7600x-1600y+20=0,-1000x-6000y+20=0$$

Kōwhiria tētahi o ngā whārite ka whakaotia mō te x mā te wehe i te x i te taha mauī o te tohu ōrite.

$$-7600x-1600y+20=0$$

Me tango 20 mai i ngā taha e rua o te whārite.

$$-7600x-1600y=-20$$

Me tāpiri 1600y ki ngā taha e rua o te whārite.

$$-7600x=1600y-20$$

Whakawehea ngā taha e rua ki te -7600.

$$x=-\frac{1}{7600}\left(1600y-20\right)$$

Whakareatia -\frac{1}{7600} ki te 1600y-20.

$$x=-\frac{4}{19}y+\frac{1}{380}$$

Whakakapia te -\frac{4y}{19}+\frac{1}{380} mō te x ki tērā atu whārite, -1000x-6000y+20=0.

$$-1000\left(-\frac{4}{19}y+\frac{1}{380}\right)-6000y+20=0$$

Whakareatia -1000 ki te -\frac{4y}{19}+\frac{1}{380}.

$$\frac{4000}{19}y-\frac{50}{19}-6000y+20=0$$

Tāpiri \frac{4000y}{19} ki te -6000y.

$$-\frac{110000}{19}y-\frac{50}{19}+20=0$$

Tāpiri -\frac{50}{19} ki te 20.

$$-\frac{110000}{19}y+\frac{330}{19}=0$$

Me tango \frac{330}{19} mai i ngā taha e rua o te whārite.

$$-\frac{110000}{19}y=-\frac{330}{19}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{110000}{19}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{3}{1000}$$

Whakaurua te \frac{3}{1000} mō y ki x=-\frac{4}{19}y+\frac{1}{380}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{4}{19}\times \frac{3}{1000}+\frac{1}{380}$$

Whakareatia -\frac{4}{19} ki te \frac{3}{1000} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=-\frac{3}{4750}+\frac{1}{380}$$

Tāpiri \frac{1}{380} ki te -\frac{3}{4750} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=\frac{1}{500}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{1}{500},y=\frac{3}{1000}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$-7600x-1600y+20=0,-1000x-6000y+20=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-7600&-1600\\-1000&-6000\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-20\\-20\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-7600&-1600\\-1000&-6000\end{matrix}\right).

$$inverse(\left(\begin{matrix}-7600&-1600\\-1000&-6000\end{matrix}\right))\left(\begin{matrix}-7600&-1600\\-1000&-6000\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-7600&-1600\\-1000&-6000\end{matrix}\right))\left(\begin{matrix}-20\\-20\end{matrix}\right)$$

Ko te hua o tētahi poukapa me te kōaro ko te poukapa tuakiri.

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-7600&-1600\\-1000&-6000\end{matrix}\right))\left(\begin{matrix}-20\\-20\end{matrix}\right)$$

Whakareatia ngā poukapa kei te taha mauī o te tohu ōrite.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-7600&-1600\\-1000&-6000\end{matrix}\right))\left(\begin{matrix}-20\\-20\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right) te poukapa kōaro, nō reira ka taea te tuhi anō te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{6000}{-7600\left(-6000\right)-\left(-1600\left(-1000\right)\right)}&-\frac{-1600}{-7600\left(-6000\right)-\left(-1600\left(-1000\right)\right)}\\-\frac{-1000}{-7600\left(-6000\right)-\left(-1600\left(-1000\right)\right)}&-\frac{7600}{-7600\left(-6000\right)-\left(-1600\left(-1000\right)\right)}\end{matrix}\right)\left(\begin{matrix}-20\\-20\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{22000}&\frac{1}{27500}\\\frac{1}{44000}&-\frac{19}{110000}\end{matrix}\right)\left(\begin{matrix}-20\\-20\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{22000}\left(-20\right)+\frac{1}{27500}\left(-20\right)\\\frac{1}{44000}\left(-20\right)-\frac{19}{110000}\left(-20\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{500}\\\frac{3}{1000}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{1}{500},y=\frac{3}{1000}$$

Ngā Upane e Whakamahi i te Tangohanga

Hei whakaoti mā te tangohanga, ko ngā tau whakarea o tētahi o ngā taurangi me mātua ōrite i ngā whārite e rua kia whakakorehia ai te taurangi ina tangohia tētahi whārite mai i tētahi atu.

$$-7600x-1600y+20=0,-1000x-6000y+20=0$$

Kia ōrite ai a -7600x me -1000x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -1000 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te -7600.

$$-1000\left(-7600\right)x-1000\left(-1600\right)y-1000\times 20=0,-7600\left(-1000\right)x-7600\left(-6000\right)y-7600\times 20=0$$

Whakarūnātia.

$$7600000x+1600000y-20000=0,7600000x+45600000y-152000=0$$

Me tango 7600000x+45600000y-152000=0 mai i 7600000x+1600000y-20000=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$7600000x-7600000x+1600000y-45600000y-20000+152000=0$$

Tāpiri 7600000x ki te -7600000x. Ka whakakore atu ngā kupu 7600000x me -7600000x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$1600000y-45600000y-20000+152000=0$$

Tāpiri 1600000y ki te -45600000y.

$$-44000000y-20000+152000=0$$

Tāpiri -20000 ki te 152000.

$$-44000000y+132000=0$$

Me tango 132000 mai i ngā taha e rua o te whārite.

$$-44000000y=-132000$$

Whakawehea ngā taha e rua ki te -44000000.

$$y=\frac{3}{1000}$$

Whakaurua te \frac{3}{1000} mō y ki -1000x-6000y+20=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-1000x-6000\times \frac{3}{1000}+20=0$$

Whakareatia -6000 ki te \frac{3}{1000}.

$$-1000x-18+20=0$$

Tāpiri -18 ki te 20.

$$-1000x+2=0$$

Me tango 2 mai i ngā taha e rua o te whārite.

$$-1000x=-2$$

Whakawehea ngā taha e rua ki te -1000.

$$x=\frac{1}{500}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{1}{500},y=\frac{3}{1000}$$