Whakaoti i te {l}{x+3y=4}{-2x+y=-22}

$$\left. \begin{array} { l } { x + 3 y = 4 } \\ { - 2 x + y = - 22 } \end{array} \right.$$

Whakaoti mō x, y

$$x=10$$

$$y=-2$$

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.

$$x+3y=4,-2x+y=-22$$

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

$$x+3y=4$$

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

$$x=-3y+4$$

Whakakapia te -3y+4 mō te x ki tērā atu whārite, -2x+y=-22.

$$-2\left(-3y+4\right)+y=-22$$

Whakareatia -2 ki te -3y+4.

$$6y-8+y=-22$$

Tāpiri 6y ki te y.

$$7y-8=-22$$

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

$$7y=-14$$

Whakawehea ngā taha e rua ki te 7.

$$y=-2$$

Whakaurua te -2 mō y ki x=-3y+4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-3\left(-2\right)+4$$

Whakareatia -3 ki te -2.

$$x=6+4$$

Tāpiri 4 ki te 6.

$$x=10$$

Kua oti te pūnaha te whakatau.

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

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.

$$x+3y=4,-2x+y=-22$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&3\\-2&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\\-22\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&3\\-2&1\end{matrix}\right))\left(\begin{matrix}1&3\\-2&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&3\\-2&1\end{matrix}\right))\left(\begin{matrix}4\\-22\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&3\\-2&1\end{matrix}\right))\left(\begin{matrix}4\\-22\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&3\\-2&1\end{matrix}\right))\left(\begin{matrix}4\\-22\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{1-3\left(-2\right)}&-\frac{3}{1-3\left(-2\right)}\\-\frac{-2}{1-3\left(-2\right)}&\frac{1}{1-3\left(-2\right)}\end{matrix}\right)\left(\begin{matrix}4\\-22\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{7}&-\frac{3}{7}\\\frac{2}{7}&\frac{1}{7}\end{matrix}\right)\left(\begin{matrix}4\\-22\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{7}\times 4-\frac{3}{7}\left(-22\right)\\\frac{2}{7}\times 4+\frac{1}{7}\left(-22\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

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

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.

$$x+3y=4,-2x+y=-22$$

Kia ōrite ai a x me -2x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$-2x-2\times 3y=-2\times 4,-2x+y=-22$$

Whakarūnātia.

$$-2x-6y=-8,-2x+y=-22$$

Me tango -2x+y=-22 mai i -2x-6y=-8 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-2x+2x-6y-y=-8+22$$

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

$$-6y-y=-8+22$$

Tāpiri -6y ki te -y.

$$-7y=-8+22$$

Tāpiri -8 ki te 22.

$$-7y=14$$

Whakawehea ngā taha e rua ki te -7.

$$y=-2$$

Whakaurua te -2 mō y ki -2x+y=-22. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-2x-2=-22$$

Me tāpiri 2 ki ngā taha e rua o te whārite.

$$-2x=-20$$

Whakawehea ngā taha e rua ki te -2.

$$x=10$$

Kua oti te pūnaha te whakatau.

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