Whakaoti i te {l}{8x+9y=3}{x+y=0}

$$\left. \begin{array}{l}{ 8 x + 9 y = 3 }\\{ x + y = 0 }\end{array} \right.$$

Whakaoti mō x, y

$$x=-3$$

$$y=3$$

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.

$$8x+9y=3,x+y=0$$

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

$$8x+9y=3$$

Me tango 9y mai i ngā taha e rua o te whārite.

$$8x=-9y+3$$

Whakawehea ngā taha e rua ki te 8.

$$x=\frac{1}{8}\left(-9y+3\right)$$

Whakareatia \frac{1}{8} ki te -9y+3.

$$x=-\frac{9}{8}y+\frac{3}{8}$$

Whakakapia te \frac{-9y+3}{8} mō te x ki tērā atu whārite, x+y=0.

$$-\frac{9}{8}y+\frac{3}{8}+y=0$$

Tāpiri -\frac{9y}{8} ki te y.

$$-\frac{1}{8}y+\frac{3}{8}=0$$

Me tango \frac{3}{8} mai i ngā taha e rua o te whārite.

$$-\frac{1}{8}y=-\frac{3}{8}$$

Me whakarea ngā taha e rua ki te -8.

$$y=3$$

Whakaurua te 3 mō y ki x=-\frac{9}{8}y+\frac{3}{8}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{9}{8}\times 3+\frac{3}{8}$$

Whakareatia -\frac{9}{8} ki te 3.

$$x=\frac{-27+3}{8}$$

Tāpiri \frac{3}{8} ki te -\frac{27}{8} 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.

$$x=-3$$

Kua oti te pūnaha te whakatau.

$$x=-3,y=3$$

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.

$$8x+9y=3,x+y=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}8&9\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3\\0\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}8&9\\1&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}8&9\\1&1\end{matrix}\right))\left(\begin{matrix}8&9\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&9\\1&1\end{matrix}\right))\left(\begin{matrix}3\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&9\\1&1\end{matrix}\right))\left(\begin{matrix}3\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&9\\1&1\end{matrix}\right))\left(\begin{matrix}3\\0\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{8-9}&-\frac{9}{8-9}\\-\frac{1}{8-9}&\frac{8}{8-9}\end{matrix}\right)\left(\begin{matrix}3\\0\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-1&9\\1&-8\end{matrix}\right)\left(\begin{matrix}3\\0\end{matrix}\right)$$

Whakareatia ngā poukapa.

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

Tangohia ngā huānga poukapa x me y.

$$x=-3,y=3$$

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.

$$8x+9y=3,x+y=0$$

Kia ōrite ai a 8x 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 8.

$$8x+9y=3,8x+8y=0$$

Me tango 8x+8y=0 mai i 8x+9y=3 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$8x-8x+9y-8y=3$$

Tāpiri 8x ki te -8x. Ka whakakore atu ngā kupu 8x me -8x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$9y-8y=3$$

Tāpiri 9y ki te -8y.

$$y=3$$

Whakaurua te 3 mō y ki x+y=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x+3=0$$

Me tango 3 mai i ngā taha e rua o te whārite.

$$x=-3$$

Kua oti te pūnaha te whakatau.

$$x=-3,y=3$$