Whakaoti i te {r}{10x-10y=-10}{+-10x+8y=12}

$$\left. \begin{array} { r } { 10 x - 10 y = - 10 } \\ { + - 10 x + 8 y = 12 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-2$$

$$y=-1$$

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.

$$10x-10y=-10,-10x+8y=12$$

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

$$10x-10y=-10$$

Me tāpiri 10y ki ngā taha e rua o te whārite.

$$10x=10y-10$$

Whakawehea ngā taha e rua ki te 10.

$$x=\frac{1}{10}\left(10y-10\right)$$

Whakareatia \frac{1}{10} ki te -10+10y.

$$x=y-1$$

Whakakapia te y-1 mō te x ki tērā atu whārite, -10x+8y=12.

$$-10\left(y-1\right)+8y=12$$

Whakareatia -10 ki te y-1.

$$-10y+10+8y=12$$

Tāpiri -10y ki te 8y.

$$-2y+10=12$$

Me tango 10 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=y-1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-1-1$$

Tāpiri -1 ki te -1.

$$x=-2$$

Kua oti te pūnaha te whakatau.

$$x=-2,y=-1$$

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.

$$10x-10y=-10,-10x+8y=12$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}10&-10\\-10&8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-10\\12\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}10&-10\\-10&8\end{matrix}\right).

$$inverse(\left(\begin{matrix}10&-10\\-10&8\end{matrix}\right))\left(\begin{matrix}10&-10\\-10&8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}10&-10\\-10&8\end{matrix}\right))\left(\begin{matrix}-10\\12\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}10&-10\\-10&8\end{matrix}\right))\left(\begin{matrix}-10\\12\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}10&-10\\-10&8\end{matrix}\right))\left(\begin{matrix}-10\\12\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{8}{10\times 8-\left(-10\left(-10\right)\right)}&-\frac{-10}{10\times 8-\left(-10\left(-10\right)\right)}\\-\frac{-10}{10\times 8-\left(-10\left(-10\right)\right)}&\frac{10}{10\times 8-\left(-10\left(-10\right)\right)}\end{matrix}\right)\left(\begin{matrix}-10\\12\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{5}&-\frac{1}{2}\\-\frac{1}{2}&-\frac{1}{2}\end{matrix}\right)\left(\begin{matrix}-10\\12\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{5}\left(-10\right)-\frac{1}{2}\times 12\\-\frac{1}{2}\left(-10\right)-\frac{1}{2}\times 12\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=-2,y=-1$$

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.

$$10x-10y=-10,-10x+8y=12$$

Kia ōrite ai a 10x me -10x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -10 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 10.

$$-10\times 10x-10\left(-10\right)y=-10\left(-10\right),10\left(-10\right)x+10\times 8y=10\times 12$$

Whakarūnātia.

$$-100x+100y=100,-100x+80y=120$$

Me tango -100x+80y=120 mai i -100x+100y=100 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-100x+100x+100y-80y=100-120$$

Tāpiri -100x ki te 100x. Ka whakakore atu ngā kupu -100x me 100x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$100y-80y=100-120$$

Tāpiri 100y ki te -80y.

$$20y=100-120$$

Tāpiri 100 ki te -120.

$$20y=-20$$

Whakawehea ngā taha e rua ki te 20.

$$y=-1$$

Whakaurua te -1 mō y ki -10x+8y=12. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-10x+8\left(-1\right)=12$$

Whakareatia 8 ki te -1.

$$-10x-8=12$$

Me tāpiri 8 ki ngā taha e rua o te whārite.

$$-10x=20$$

Whakawehea ngā taha e rua ki te -10.

$$x=-2$$

Kua oti te pūnaha te whakatau.

$$x=-2,y=-1$$