Whakaoti i te {l}{9x+y=10}{x+3y=-3}

$$\left\{ \begin{array} { l } { 9 x + y = 10 } \\ { x + 3 y = - 3 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{33}{26} = 1\frac{7}{26} \approx 1.269230769$$

$$y = -\frac{37}{26} = -1\frac{11}{26} \approx -1.423076923$$

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.

$$9x+y=10,x+3y=-3$$

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

$$9x+y=10$$

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

$$9x=-y+10$$

Whakawehea ngā taha e rua ki te 9.

$$x=\frac{1}{9}\left(-y+10\right)$$

Whakareatia \frac{1}{9} ki te -y+10.

$$x=-\frac{1}{9}y+\frac{10}{9}$$

Whakakapia te \frac{-y+10}{9} mō te x ki tērā atu whārite, x+3y=-3.

$$-\frac{1}{9}y+\frac{10}{9}+3y=-3$$

Tāpiri -\frac{y}{9} ki te 3y.

$$\frac{26}{9}y+\frac{10}{9}=-3$$

Me tango \frac{10}{9} mai i ngā taha e rua o te whārite.

$$\frac{26}{9}y=-\frac{37}{9}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{26}{9}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-\frac{37}{26}$$

Whakaurua te -\frac{37}{26} mō y ki x=-\frac{1}{9}y+\frac{10}{9}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{9}\left(-\frac{37}{26}\right)+\frac{10}{9}$$

Whakareatia -\frac{1}{9} ki te -\frac{37}{26} 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.

$$x=\frac{37}{234}+\frac{10}{9}$$

Tāpiri \frac{10}{9} ki te \frac{37}{234} 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.

$$x=\frac{33}{26}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{33}{26},y=-\frac{37}{26}$$

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.

$$9x+y=10,x+3y=-3$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}9&1\\1&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}10\\-3\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}9&1\\1&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}9&1\\1&3\end{matrix}\right))\left(\begin{matrix}9&1\\1&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}9&1\\1&3\end{matrix}\right))\left(\begin{matrix}10\\-3\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}9&1\\1&3\end{matrix}\right))\left(\begin{matrix}10\\-3\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}9&1\\1&3\end{matrix}\right))\left(\begin{matrix}10\\-3\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{9\times 3-1}&-\frac{1}{9\times 3-1}\\-\frac{1}{9\times 3-1}&\frac{9}{9\times 3-1}\end{matrix}\right)\left(\begin{matrix}10\\-3\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{26}&-\frac{1}{26}\\-\frac{1}{26}&\frac{9}{26}\end{matrix}\right)\left(\begin{matrix}10\\-3\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{26}\times 10-\frac{1}{26}\left(-3\right)\\-\frac{1}{26}\times 10+\frac{9}{26}\left(-3\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{33}{26}\\-\frac{37}{26}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{33}{26},y=-\frac{37}{26}$$

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.

$$9x+y=10,x+3y=-3$$

Kia ōrite ai a 9x me x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 1 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 9.

$$9x+y=10,9x+9\times 3y=9\left(-3\right)$$

Whakarūnātia.

$$9x+y=10,9x+27y=-27$$

Me tango 9x+27y=-27 mai i 9x+y=10 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$9x-9x+y-27y=10+27$$

Tāpiri 9x ki te -9x. Ka whakakore atu ngā kupu 9x me -9x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$y-27y=10+27$$

Tāpiri y ki te -27y.

$$-26y=10+27$$

Tāpiri 10 ki te 27.

$$-26y=37$$

Whakawehea ngā taha e rua ki te -26.

$$y=-\frac{37}{26}$$

Whakaurua te -\frac{37}{26} mō y ki x+3y=-3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x+3\left(-\frac{37}{26}\right)=-3$$

Whakareatia 3 ki te -\frac{37}{26}.

$$x-\frac{111}{26}=-3$$

Me tāpiri \frac{111}{26} ki ngā taha e rua o te whārite.

$$x=\frac{33}{26}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{33}{26},y=-\frac{37}{26}$$