Whakaoti i te {l}{2x+8y=16}{11-x+2y=0}

$$\left\{ \begin{array} { l } { 2 x + 8 y = 16 } \\ { 11 - x + 2 y = 0 } \end{array} \right.$$

Whakaoti mō x, y

$$x=10$$

$$y=-\frac{1}{2}=-0.5$$

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.

$$2x+8y=16,-x+2y+11=0$$

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

$$2x+8y=16$$

Me tango 8y mai i ngā taha e rua o te whārite.

$$2x=-8y+16$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(-8y+16\right)$$

Whakareatia \frac{1}{2} ki te -8y+16.

$$x=-4y+8$$

Whakakapia te -4y+8 mō te x ki tērā atu whārite, -x+2y+11=0.

$$-\left(-4y+8\right)+2y+11=0$$

Whakareatia -1 ki te -4y+8.

$$4y-8+2y+11=0$$

Tāpiri 4y ki te 2y.

$$6y-8+11=0$$

Tāpiri -8 ki te 11.

$$6y+3=0$$

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

$$6y=-3$$

Whakawehea ngā taha e rua ki te 6.

$$y=-\frac{1}{2}$$

Whakaurua te -\frac{1}{2} mō y ki x=-4y+8. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-4\left(-\frac{1}{2}\right)+8$$

Whakareatia -4 ki te -\frac{1}{2}.

$$x=2+8$$

Tāpiri 8 ki te 2.

$$x=10$$

Kua oti te pūnaha te whakatau.

$$x=10,y=-\frac{1}{2}$$

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.

$$2x+8y=16,-x+2y+11=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&8\\-1&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}16\\-11\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&8\\-1&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&8\\-1&2\end{matrix}\right))\left(\begin{matrix}2&8\\-1&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&8\\-1&2\end{matrix}\right))\left(\begin{matrix}16\\-11\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&8\\-1&2\end{matrix}\right))\left(\begin{matrix}16\\-11\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&8\\-1&2\end{matrix}\right))\left(\begin{matrix}16\\-11\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{2\times 2-8\left(-1\right)}&-\frac{8}{2\times 2-8\left(-1\right)}\\-\frac{-1}{2\times 2-8\left(-1\right)}&\frac{2}{2\times 2-8\left(-1\right)}\end{matrix}\right)\left(\begin{matrix}16\\-11\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{6}&-\frac{2}{3}\\\frac{1}{12}&\frac{1}{6}\end{matrix}\right)\left(\begin{matrix}16\\-11\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{6}\times 16-\frac{2}{3}\left(-11\right)\\\frac{1}{12}\times 16+\frac{1}{6}\left(-11\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}10\\-\frac{1}{2}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=10,y=-\frac{1}{2}$$

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.

$$2x+8y=16,-x+2y+11=0$$

Kia ōrite ai a 2x 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 2.

$$-2x-8y=-16,2\left(-1\right)x+2\times 2y+2\times 11=0$$

Whakarūnātia.

$$-2x-8y=-16,-2x+4y+22=0$$

Me tango -2x+4y+22=0 mai i -2x-8y=-16 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-2x+2x-8y-4y-22=-16$$

Tāpiri -2x ki te 2x. Ka whakakore atu ngā kupu -2x me 2x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-8y-4y-22=-16$$

Tāpiri -8y ki te -4y.

$$-12y-22=-16$$

Me tāpiri 22 ki ngā taha e rua o te whārite.

$$-12y=6$$

Whakawehea ngā taha e rua ki te -12.

$$y=-\frac{1}{2}$$

Whakaurua te -\frac{1}{2} mō y ki -x+2y+11=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-x+2\left(-\frac{1}{2}\right)+11=0$$

Whakareatia 2 ki te -\frac{1}{2}.

$$-x-1+11=0$$

Tāpiri -1 ki te 11.

$$-x+10=0$$

Me tango 10 mai i ngā taha e rua o te whārite.

$$-x=-10$$

Whakawehea ngā taha e rua ki te -1.

$$x=10$$

Kua oti te pūnaha te whakatau.

$$x=10,y=-\frac{1}{2}$$