Whakaoti i te {l}{9x+71y=135}{4x+y=41}

$$\left\{ \begin{array} { l } { 9 x + 71 y = 135 } \\ { 4 x + y = 41 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{2776}{275} = 10\frac{26}{275} \approx 10.094545455$$

$$y=\frac{171}{275}\approx 0.621818182$$

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.

$$9x+71y=135,4x+y=41$$

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

$$9x+71y=135$$

Me tango 71y mai i ngā taha e rua o te whārite.

$$9x=-71y+135$$

Whakawehea ngā taha e rua ki te 9.

$$x=\frac{1}{9}\left(-71y+135\right)$$

Whakareatia \frac{1}{9} ki te -71y+135.

$$x=-\frac{71}{9}y+15$$

Whakakapia te -\frac{71y}{9}+15 mō te x ki tērā atu whārite, 4x+y=41.

$$4\left(-\frac{71}{9}y+15\right)+y=41$$

Whakareatia 4 ki te -\frac{71y}{9}+15.

$$-\frac{284}{9}y+60+y=41$$

Tāpiri -\frac{284y}{9} ki te y.

$$-\frac{275}{9}y+60=41$$

Me tango 60 mai i ngā taha e rua o te whārite.

$$-\frac{275}{9}y=-19$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{275}{9}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{171}{275}$$

Whakaurua te \frac{171}{275} mō y ki x=-\frac{71}{9}y+15. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{71}{9}\times \frac{171}{275}+15$$

Whakareatia -\frac{71}{9} ki te \frac{171}{275} 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.

$$x=-\frac{1349}{275}+15$$

Tāpiri 15 ki te -\frac{1349}{275}.

$$x=\frac{2776}{275}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{2776}{275},y=\frac{171}{275}$$

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.

$$9x+71y=135,4x+y=41$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}9&71\\4&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}135\\41\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}9&71\\4&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}9&71\\4&1\end{matrix}\right))\left(\begin{matrix}9&71\\4&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}9&71\\4&1\end{matrix}\right))\left(\begin{matrix}135\\41\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}9&71\\4&1\end{matrix}\right))\left(\begin{matrix}135\\41\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}9&71\\4&1\end{matrix}\right))\left(\begin{matrix}135\\41\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{9-71\times 4}&-\frac{71}{9-71\times 4}\\-\frac{4}{9-71\times 4}&\frac{9}{9-71\times 4}\end{matrix}\right)\left(\begin{matrix}135\\41\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{275}&\frac{71}{275}\\\frac{4}{275}&-\frac{9}{275}\end{matrix}\right)\left(\begin{matrix}135\\41\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{275}\times 135+\frac{71}{275}\times 41\\\frac{4}{275}\times 135-\frac{9}{275}\times 41\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2776}{275}\\\frac{171}{275}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{2776}{275},y=\frac{171}{275}$$

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.

$$9x+71y=135,4x+y=41$$

Kia ōrite ai a 9x me 4x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 4 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 9.

$$4\times 9x+4\times 71y=4\times 135,9\times 4x+9y=9\times 41$$

Whakarūnātia.

$$36x+284y=540,36x+9y=369$$

Me tango 36x+9y=369 mai i 36x+284y=540 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$36x-36x+284y-9y=540-369$$

Tāpiri 36x ki te -36x. Ka whakakore atu ngā kupu 36x me -36x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$284y-9y=540-369$$

Tāpiri 284y ki te -9y.

$$275y=540-369$$

Tāpiri 540 ki te -369.

$$275y=171$$

Whakawehea ngā taha e rua ki te 275.

$$y=\frac{171}{275}$$

Whakaurua te \frac{171}{275} mō y ki 4x+y=41. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x+\frac{171}{275}=41$$

Me tango \frac{171}{275} mai i ngā taha e rua o te whārite.

$$4x=\frac{11104}{275}$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{2776}{275}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{2776}{275},y=\frac{171}{275}$$