Whakaoti i te {r}{2x+3y=10}{-3x+y=18}

$$\left. \begin{array} { r } { 2 x + 3 y = 10 } \\ { - 3 x + y = 18 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-4$$

$$y=6$$

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.

$$2x+3y=10,-3x+y=18$$

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

$$2x+3y=10$$

Me tango 3y mai i ngā taha e rua o te whārite.

$$2x=-3y+10$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(-3y+10\right)$$

Whakareatia \frac{1}{2} ki te -3y+10.

$$x=-\frac{3}{2}y+5$$

Whakakapia te -\frac{3y}{2}+5 mō te x ki tērā atu whārite, -3x+y=18.

$$-3\left(-\frac{3}{2}y+5\right)+y=18$$

Whakareatia -3 ki te -\frac{3y}{2}+5.

$$\frac{9}{2}y-15+y=18$$

Tāpiri \frac{9y}{2} ki te y.

$$\frac{11}{2}y-15=18$$

Me tāpiri 15 ki ngā taha e rua o te whārite.

$$\frac{11}{2}y=33$$

Whakawehea ngā taha e rua o te whārite ki te \frac{11}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=6$$

Whakaurua te 6 mō y ki x=-\frac{3}{2}y+5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{3}{2}\times 6+5$$

Whakareatia -\frac{3}{2} ki te 6.

$$x=-9+5$$

Tāpiri 5 ki te -9.

$$x=-4$$

Kua oti te pūnaha te whakatau.

$$x=-4,y=6$$

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.

$$2x+3y=10,-3x+y=18$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&3\\-3&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}10\\18\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&3\\-3&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&3\\-3&1\end{matrix}\right))\left(\begin{matrix}2&3\\-3&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\-3&1\end{matrix}\right))\left(\begin{matrix}10\\18\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\-3&1\end{matrix}\right))\left(\begin{matrix}10\\18\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\-3&1\end{matrix}\right))\left(\begin{matrix}10\\18\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te poukapa kōaro ko \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), kia tuhia anō ai te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2-3\left(-3\right)}&-\frac{3}{2-3\left(-3\right)}\\-\frac{-3}{2-3\left(-3\right)}&\frac{2}{2-3\left(-3\right)}\end{matrix}\right)\left(\begin{matrix}10\\18\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{11}&-\frac{3}{11}\\\frac{3}{11}&\frac{2}{11}\end{matrix}\right)\left(\begin{matrix}10\\18\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{11}\times 10-\frac{3}{11}\times 18\\\frac{3}{11}\times 10+\frac{2}{11}\times 18\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-4\\6\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-4,y=6$$

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.

$$2x+3y=10,-3x+y=18$$

Kia ōrite ai a 2x me -3x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -3 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

$$-3\times 2x-3\times 3y=-3\times 10,2\left(-3\right)x+2y=2\times 18$$

Whakarūnātia.

$$-6x-9y=-30,-6x+2y=36$$

Me tango -6x+2y=36 mai i -6x-9y=-30 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-6x+6x-9y-2y=-30-36$$

Tāpiri -6x ki te 6x. Ka whakakore atu ngā kupu -6x me 6x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-9y-2y=-30-36$$

Tāpiri -9y ki te -2y.

$$-11y=-30-36$$

Tāpiri -30 ki te -36.

$$-11y=-66$$

Whakawehea ngā taha e rua ki te -11.

$$y=6$$

Whakaurua te 6 mō y ki -3x+y=18. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-3x+6=18$$

Me tango 6 mai i ngā taha e rua o te whārite.

$$-3x=12$$

Whakawehea ngā taha e rua ki te -3.

$$x=-4$$

Kua oti te pūnaha te whakatau.

$$x=-4,y=6$$