Whakaoti i te {l}{x-y=-2}{11x-4y=-36}

$$\left. \begin{array} { l } { x - y = - 2 } \\ { 11 x - 4 y = - 36 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-4$$

$$y=-2$$

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.

$$x-y=-2,11x-4y=-36$$

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

$$x-y=-2$$

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

$$x=y-2$$

Whakakapia te y-2 mō te x ki tērā atu whārite, 11x-4y=-36.

$$11\left(y-2\right)-4y=-36$$

Whakareatia 11 ki te y-2.

$$11y-22-4y=-36$$

Tāpiri 11y ki te -4y.

$$7y-22=-36$$

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

$$7y=-14$$

Whakawehea ngā taha e rua ki te 7.

$$y=-2$$

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

$$x=-2-2$$

Tāpiri -2 ki te -2.

$$x=-4$$

Kua oti te pūnaha te whakatau.

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

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.

$$x-y=-2,11x-4y=-36$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-1\\11&-4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-2\\-36\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-1\\11&-4\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-1\\11&-4\end{matrix}\right))\left(\begin{matrix}1&-1\\11&-4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\11&-4\end{matrix}\right))\left(\begin{matrix}-2\\-36\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\11&-4\end{matrix}\right))\left(\begin{matrix}-2\\-36\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\11&-4\end{matrix}\right))\left(\begin{matrix}-2\\-36\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{-4-\left(-11\right)}&-\frac{-1}{-4-\left(-11\right)}\\-\frac{11}{-4-\left(-11\right)}&\frac{1}{-4-\left(-11\right)}\end{matrix}\right)\left(\begin{matrix}-2\\-36\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{7}&\frac{1}{7}\\-\frac{11}{7}&\frac{1}{7}\end{matrix}\right)\left(\begin{matrix}-2\\-36\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{7}\left(-2\right)+\frac{1}{7}\left(-36\right)\\-\frac{11}{7}\left(-2\right)+\frac{1}{7}\left(-36\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

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

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.

$$x-y=-2,11x-4y=-36$$

Kia ōrite ai a x me 11x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 11 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$11x+11\left(-1\right)y=11\left(-2\right),11x-4y=-36$$

Whakarūnātia.

$$11x-11y=-22,11x-4y=-36$$

Me tango 11x-4y=-36 mai i 11x-11y=-22 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$11x-11x-11y+4y=-22+36$$

Tāpiri 11x ki te -11x. Ka whakakore atu ngā kupu 11x me -11x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-11y+4y=-22+36$$

Tāpiri -11y ki te 4y.

$$-7y=-22+36$$

Tāpiri -22 ki te 36.

$$-7y=14$$

Whakawehea ngā taha e rua ki te -7.

$$y=-2$$

Whakaurua te -2 mō y ki 11x-4y=-36. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$11x-4\left(-2\right)=-36$$

Whakareatia -4 ki te -2.

$$11x+8=-36$$

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

$$11x=-44$$

Whakawehea ngā taha e rua ki te 11.

$$x=-4$$

Kua oti te pūnaha te whakatau.

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