Whakaoti i te {l}{x-10y=-14}{-5x-8y=12}

$$\left. \begin{array} { l } { x - 10 y = - 14 } \\ { - 5 x - 8 y = 12 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-4$$

$$y=1$$

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.

$$x-10y=-14,-5x-8y=12$$

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

$$x-10y=-14$$

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

$$x=10y-14$$

Whakakapia te 10y-14 mō te x ki tērā atu whārite, -5x-8y=12.

$$-5\left(10y-14\right)-8y=12$$

Whakareatia -5 ki te 10y-14.

$$-50y+70-8y=12$$

Tāpiri -50y ki te -8y.

$$-58y+70=12$$

Me tango 70 mai i ngā taha e rua o te whārite.

$$-58y=-58$$

Whakawehea ngā taha e rua ki te -58.

$$y=1$$

Whakaurua te 1 mō y ki x=10y-14. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=10-14$$

Tāpiri -14 ki te 10.

$$x=-4$$

Kua oti te pūnaha te whakatau.

$$x=-4,y=1$$

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.

$$x-10y=-14,-5x-8y=12$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-10\\-5&-8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-14\\12\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-10\\-5&-8\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-10\\-5&-8\end{matrix}\right))\left(\begin{matrix}1&-10\\-5&-8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-10\\-5&-8\end{matrix}\right))\left(\begin{matrix}-14\\12\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-10\\-5&-8\end{matrix}\right))\left(\begin{matrix}-14\\12\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-10\\-5&-8\end{matrix}\right))\left(\begin{matrix}-14\\12\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{8}{-8-\left(-10\left(-5\right)\right)}&-\frac{-10}{-8-\left(-10\left(-5\right)\right)}\\-\frac{-5}{-8-\left(-10\left(-5\right)\right)}&\frac{1}{-8-\left(-10\left(-5\right)\right)}\end{matrix}\right)\left(\begin{matrix}-14\\12\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{29}&-\frac{5}{29}\\-\frac{5}{58}&-\frac{1}{58}\end{matrix}\right)\left(\begin{matrix}-14\\12\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{29}\left(-14\right)-\frac{5}{29}\times 12\\-\frac{5}{58}\left(-14\right)-\frac{1}{58}\times 12\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=-4,y=1$$

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.

$$x-10y=-14,-5x-8y=12$$

Kia ōrite ai a x me -5x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -5 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$-5x-5\left(-10\right)y=-5\left(-14\right),-5x-8y=12$$

Whakarūnātia.

$$-5x+50y=70,-5x-8y=12$$

Me tango -5x-8y=12 mai i -5x+50y=70 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-5x+5x+50y+8y=70-12$$

Tāpiri -5x ki te 5x. Ka whakakore atu ngā kupu -5x me 5x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$50y+8y=70-12$$

Tāpiri 50y ki te 8y.

$$58y=70-12$$

Tāpiri 70 ki te -12.

$$58y=58$$

Whakawehea ngā taha e rua ki te 58.

$$y=1$$

Whakaurua te 1 mō y ki -5x-8y=12. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-5x-8=12$$

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

$$-5x=20$$

Whakawehea ngā taha e rua ki te -5.

$$x=-4$$

Kua oti te pūnaha te whakatau.

$$x=-4,y=1$$