Whakaoti i te {l}{-x-2y=-7}{2x+2y=16}

$$\left. \begin{array} { l } { - x - 2 y = - 7 } \\ { 2 x + 2 y = 16 } \end{array} \right.$$

Whakaoti mō x, y

$$x=9$$

$$y=-1$$

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.

$$-x-2y=-7,2x+2y=16$$

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

$$-x-2y=-7$$

Me tāpiri 2y ki ngā taha e rua o te whārite.

$$-x=2y-7$$

Whakawehea ngā taha e rua ki te -1.

$$x=-\left(2y-7\right)$$

Whakareatia -1 ki te 2y-7.

$$x=-2y+7$$

Whakakapia te -2y+7 mō te x ki tērā atu whārite, 2x+2y=16.

$$2\left(-2y+7\right)+2y=16$$

Whakareatia 2 ki te -2y+7.

$$-4y+14+2y=16$$

Tāpiri -4y ki te 2y.

$$-2y+14=16$$

Me tango 14 mai i ngā taha e rua o te whārite.

$$-2y=2$$

Whakawehea ngā taha e rua ki te -2.

$$y=-1$$

Whakaurua te -1 mō y ki x=-2y+7. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-2\left(-1\right)+7$$

Whakareatia -2 ki te -1.

$$x=2+7$$

Tāpiri 7 ki te 2.

$$x=9$$

Kua oti te pūnaha te whakatau.

$$x=9,y=-1$$

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.

$$-x-2y=-7,2x+2y=16$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-1&-2\\2&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-7\\16\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}-1&-2\\2&2\end{matrix}\right))\left(\begin{matrix}-1&-2\\2&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&-2\\2&2\end{matrix}\right))\left(\begin{matrix}-7\\16\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&-2\\2&2\end{matrix}\right))\left(\begin{matrix}-7\\16\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&-2\\2&2\end{matrix}\right))\left(\begin{matrix}-7\\16\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{-2-\left(-2\times 2\right)}&-\frac{-2}{-2-\left(-2\times 2\right)}\\-\frac{2}{-2-\left(-2\times 2\right)}&-\frac{1}{-2-\left(-2\times 2\right)}\end{matrix}\right)\left(\begin{matrix}-7\\16\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1&1\\-1&-\frac{1}{2}\end{matrix}\right)\left(\begin{matrix}-7\\16\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-7+16\\-\left(-7\right)-\frac{1}{2}\times 16\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=9,y=-1$$

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.

$$-x-2y=-7,2x+2y=16$$

Kia ōrite ai a -x me 2x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te -1.

$$2\left(-1\right)x+2\left(-2\right)y=2\left(-7\right),-2x-2y=-16$$

Whakarūnātia.

$$-2x-4y=-14,-2x-2y=-16$$

Me tango -2x-2y=-16 mai i -2x-4y=-14 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-2x+2x-4y+2y=-14+16$$

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.

$$-4y+2y=-14+16$$

Tāpiri -4y ki te 2y.

$$-2y=-14+16$$

Tāpiri -14 ki te 16.

$$-2y=2$$

Whakawehea ngā taha e rua ki te -2.

$$y=-1$$

Whakaurua te -1 mō y ki 2x+2y=16. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x+2\left(-1\right)=16$$

Whakareatia 2 ki te -1.

$$2x-2=16$$

Me tāpiri 2 ki ngā taha e rua o te whārite.

$$2x=18$$

Whakawehea ngā taha e rua ki te 2.

$$x=9$$

Kua oti te pūnaha te whakatau.

$$x=9,y=-1$$