Whakaoti i te {l}{10x-6y=22}{8x+y=4}

$$\left. \begin{array} { l } { 10 x - 6 y = 22 } \\ { 8 x + y = 4 } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{23}{29}\approx 0.793103448$$

$$y = -\frac{68}{29} = -2\frac{10}{29} \approx -2.344827586$$

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.

$$10x-6y=22,8x+y=4$$

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

$$10x-6y=22$$

Me tāpiri 6y ki ngā taha e rua o te whārite.

$$10x=6y+22$$

Whakawehea ngā taha e rua ki te 10.

$$x=\frac{1}{10}\left(6y+22\right)$$

Whakareatia \frac{1}{10} ki te 6y+22.

$$x=\frac{3}{5}y+\frac{11}{5}$$

Whakakapia te \frac{3y+11}{5} mō te x ki tērā atu whārite, 8x+y=4.

$$8\left(\frac{3}{5}y+\frac{11}{5}\right)+y=4$$

Whakareatia 8 ki te \frac{3y+11}{5}.

$$\frac{24}{5}y+\frac{88}{5}+y=4$$

Tāpiri \frac{24y}{5} ki te y.

$$\frac{29}{5}y+\frac{88}{5}=4$$

Me tango \frac{88}{5} mai i ngā taha e rua o te whārite.

$$\frac{29}{5}y=-\frac{68}{5}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{29}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-\frac{68}{29}$$

Whakaurua te -\frac{68}{29} mō y ki x=\frac{3}{5}y+\frac{11}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{3}{5}\left(-\frac{68}{29}\right)+\frac{11}{5}$$

Whakareatia \frac{3}{5} ki te -\frac{68}{29} 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.

$$x=-\frac{204}{145}+\frac{11}{5}$$

Tāpiri \frac{11}{5} ki te -\frac{204}{145} 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.

$$x=\frac{23}{29}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{23}{29},y=-\frac{68}{29}$$

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.

$$10x-6y=22,8x+y=4$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}10&-6\\8&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}22\\4\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}10&-6\\8&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}10&-6\\8&1\end{matrix}\right))\left(\begin{matrix}10&-6\\8&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}10&-6\\8&1\end{matrix}\right))\left(\begin{matrix}22\\4\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}10&-6\\8&1\end{matrix}\right))\left(\begin{matrix}22\\4\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}10&-6\\8&1\end{matrix}\right))\left(\begin{matrix}22\\4\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{10-\left(-6\times 8\right)}&-\frac{-6}{10-\left(-6\times 8\right)}\\-\frac{8}{10-\left(-6\times 8\right)}&\frac{10}{10-\left(-6\times 8\right)}\end{matrix}\right)\left(\begin{matrix}22\\4\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{58}&\frac{3}{29}\\-\frac{4}{29}&\frac{5}{29}\end{matrix}\right)\left(\begin{matrix}22\\4\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{58}\times 22+\frac{3}{29}\times 4\\-\frac{4}{29}\times 22+\frac{5}{29}\times 4\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{23}{29}\\-\frac{68}{29}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{23}{29},y=-\frac{68}{29}$$

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.

$$10x-6y=22,8x+y=4$$

Kia ōrite ai a 10x me 8x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 8 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 10.

$$8\times 10x+8\left(-6\right)y=8\times 22,10\times 8x+10y=10\times 4$$

Whakarūnātia.

$$80x-48y=176,80x+10y=40$$

Me tango 80x+10y=40 mai i 80x-48y=176 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$80x-80x-48y-10y=176-40$$

Tāpiri 80x ki te -80x. Ka whakakore atu ngā kupu 80x me -80x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-48y-10y=176-40$$

Tāpiri -48y ki te -10y.

$$-58y=176-40$$

Tāpiri 176 ki te -40.

$$-58y=136$$

Whakawehea ngā taha e rua ki te -58.

$$y=-\frac{68}{29}$$

Whakaurua te -\frac{68}{29} mō y ki 8x+y=4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$8x-\frac{68}{29}=4$$

Me tāpiri \frac{68}{29} ki ngā taha e rua o te whārite.

$$8x=\frac{184}{29}$$

Whakawehea ngā taha e rua ki te 8.

$$x=\frac{23}{29}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{23}{29},y=-\frac{68}{29}$$