Whakaoti i te {l}{-x+8y=18}{x-6y=-16}

$$\left. \begin{array} { l } { - x + 8 y = 18 } \\ { x - 6 y = - 16 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-10$$

$$y=1$$

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.

$$-x+8y=18,x-6y=-16$$

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

$$-x+8y=18$$

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

$$-x=-8y+18$$

Whakawehea ngā taha e rua ki te -1.

$$x=-\left(-8y+18\right)$$

Whakareatia -1 ki te -8y+18.

$$x=8y-18$$

Whakakapia te 8y-18 mō te x ki tērā atu whārite, x-6y=-16.

$$8y-18-6y=-16$$

Tāpiri 8y ki te -6y.

$$2y-18=-16$$

Me tāpiri 18 ki 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=8y-18. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=8-18$$

Tāpiri -18 ki te 8.

$$x=-10$$

Kua oti te pūnaha te whakatau.

$$x=-10,y=1$$

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.

$$-x+8y=18,x-6y=-16$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-1&8\\1&-6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}18\\-16\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-1&8\\1&-6\end{matrix}\right).

$$inverse(\left(\begin{matrix}-1&8\\1&-6\end{matrix}\right))\left(\begin{matrix}-1&8\\1&-6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&8\\1&-6\end{matrix}\right))\left(\begin{matrix}18\\-16\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&8\\1&-6\end{matrix}\right))\left(\begin{matrix}18\\-16\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&8\\1&-6\end{matrix}\right))\left(\begin{matrix}18\\-16\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{6}{-\left(-6\right)-8}&-\frac{8}{-\left(-6\right)-8}\\-\frac{1}{-\left(-6\right)-8}&-\frac{1}{-\left(-6\right)-8}\end{matrix}\right)\left(\begin{matrix}18\\-16\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3&4\\\frac{1}{2}&\frac{1}{2}\end{matrix}\right)\left(\begin{matrix}18\\-16\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3\times 18+4\left(-16\right)\\\frac{1}{2}\times 18+\frac{1}{2}\left(-16\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-10\\1\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-10,y=1$$

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.

$$-x+8y=18,x-6y=-16$$

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

$$-x+8y=18,-x-\left(-6y\right)=-\left(-16\right)$$

Whakarūnātia.

$$-x+8y=18,-x+6y=16$$

Me tango -x+6y=16 mai i -x+8y=18 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-x+x+8y-6y=18-16$$

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

$$8y-6y=18-16$$

Tāpiri 8y ki te -6y.

$$2y=18-16$$

Tāpiri 18 ki te -16.

$$2y=2$$

Whakawehea ngā taha e rua ki te 2.

$$y=1$$

Whakaurua te 1 mō y ki x-6y=-16. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x-6=-16$$

Me tāpiri 6 ki ngā taha e rua o te whārite.

$$x=-10$$

Kua oti te pūnaha te whakatau.

$$x=-10,y=1$$