Whakaoti i te {l}{-x+2y=17}{2x+2y=-10}

$$\left. \begin{array} { l } { - x + 2 y = 17 } \\ { 2 x + 2 y = - 10 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-9$$

$$y=4$$

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.

$$-x+2y=17,2x+2y=-10$$

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

$$-x+2y=17$$

Me tango 2y mai i ngā taha e rua o te whārite.

$$-x=-2y+17$$

Whakawehea ngā taha e rua ki te -1.

$$x=-\left(-2y+17\right)$$

Whakareatia -1 ki te -2y+17.

$$x=2y-17$$

Whakakapia te 2y-17 mō te x ki tērā atu whārite, 2x+2y=-10.

$$2\left(2y-17\right)+2y=-10$$

Whakareatia 2 ki te 2y-17.

$$4y-34+2y=-10$$

Tāpiri 4y ki te 2y.

$$6y-34=-10$$

Me tāpiri 34 ki ngā taha e rua o te whārite.

$$6y=24$$

Whakawehea ngā taha e rua ki te 6.

$$y=4$$

Whakaurua te 4 mō y ki x=2y-17. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=2\times 4-17$$

Whakareatia 2 ki te 4.

$$x=8-17$$

Tāpiri -17 ki te 8.

$$x=-9$$

Kua oti te pūnaha te whakatau.

$$x=-9,y=4$$

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.

$$-x+2y=17,2x+2y=-10$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-1&2\\2&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}17\\-10\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}-1&2\\2&2\end{matrix}\right))\left(\begin{matrix}-1&2\\2&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&2\\2&2\end{matrix}\right))\left(\begin{matrix}17\\-10\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&2\\2&2\end{matrix}\right))\left(\begin{matrix}17\\-10\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&2\\2&2\end{matrix}\right))\left(\begin{matrix}17\\-10\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{-2-2\times 2}&-\frac{2}{-2-2\times 2}\\-\frac{2}{-2-2\times 2}&-\frac{1}{-2-2\times 2}\end{matrix}\right)\left(\begin{matrix}17\\-10\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{3}&\frac{1}{3}\\\frac{1}{3}&\frac{1}{6}\end{matrix}\right)\left(\begin{matrix}17\\-10\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{3}\times 17+\frac{1}{3}\left(-10\right)\\\frac{1}{3}\times 17+\frac{1}{6}\left(-10\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-9\\4\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-9,y=4$$

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.

$$-x+2y=17,2x+2y=-10$$

Me tango 2x+2y=-10 mai i -x+2y=17 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-x-2x+2y-2y=17+10$$

Tāpiri 2y ki te -2y. Ka whakakore atu ngā kupu 2y me -2y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-x-2x=17+10$$

Tāpiri -x ki te -2x.

$$-3x=17+10$$

Tāpiri 17 ki te 10.

$$-3x=27$$

Whakawehea ngā taha e rua ki te -3.

$$x=-9$$

Whakaurua te -9 mō x ki 2x+2y=-10. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$2\left(-9\right)+2y=-10$$

Whakareatia 2 ki te -9.

$$-18+2y=-10$$

Me tāpiri 18 ki ngā taha e rua o te whārite.

$$2y=8$$

Whakawehea ngā taha e rua ki te 2.

$$y=4$$

Kua oti te pūnaha te whakatau.

$$x=-9,y=4$$