Whakaoti i te {l}{x+y=-1}{4x-2y=-16}

$$\left\{ \begin{array} { l } { x + y = - 1 } \\ { 4 x - 2 y = - 16 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-3$$

$$y=2$$

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.

$$x+y=-1,4x-2y=-16$$

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

$$x+y=-1$$

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

$$x=-y-1$$

Whakakapia te -y-1 mō te x ki tērā atu whārite, 4x-2y=-16.

$$4\left(-y-1\right)-2y=-16$$

Whakareatia 4 ki te -y-1.

$$-4y-4-2y=-16$$

Tāpiri -4y ki te -2y.

$$-6y-4=-16$$

Me tāpiri 4 ki ngā taha e rua o te whārite.

$$-6y=-12$$

Whakawehea ngā taha e rua ki te -6.

$$y=2$$

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

$$x=-2-1$$

Tāpiri -1 ki te -2.

$$x=-3$$

Kua oti te pūnaha te whakatau.

$$x=-3,y=2$$

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.

$$x+y=-1,4x-2y=-16$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\4&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-1\\-16\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&1\\4&-2\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&1\\4&-2\end{matrix}\right))\left(\begin{matrix}1&1\\4&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\4&-2\end{matrix}\right))\left(\begin{matrix}-1\\-16\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\4&-2\end{matrix}\right))\left(\begin{matrix}-1\\-16\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\4&-2\end{matrix}\right))\left(\begin{matrix}-1\\-16\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{-2-4}&-\frac{1}{-2-4}\\-\frac{4}{-2-4}&\frac{1}{-2-4}\end{matrix}\right)\left(\begin{matrix}-1\\-16\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-3\\2\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-3,y=2$$

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.

$$x+y=-1,4x-2y=-16$$

Kia ōrite ai a x 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 1.

$$4x+4y=4\left(-1\right),4x-2y=-16$$

Whakarūnātia.

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

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

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

Tāpiri 4x ki te -4x. Ka whakakore atu ngā kupu 4x me -4x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

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

Tāpiri 4y ki te 2y.

$$6y=-4+16$$

Tāpiri -4 ki te 16.

$$6y=12$$

Whakawehea ngā taha e rua ki te 6.

$$y=2$$

Whakaurua te 2 mō y ki 4x-2y=-16. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x-2\times 2=-16$$

Whakareatia -2 ki te 2.

$$4x-4=-16$$

Me tāpiri 4 ki ngā taha e rua o te whārite.

$$4x=-12$$

Whakawehea ngā taha e rua ki te 4.

$$x=-3$$

Kua oti te pūnaha te whakatau.

$$x=-3,y=2$$