Whakaoti i te {l}{3x+y=9}{2x-3y=6}

$$\left. \begin{array} { l } { 3 x + y = 9 } \\ { 2 x - 3 y = 6 } \end{array} \right.$$

Whakaoti mō x, y

$$x=3$$

$$y=0$$

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.

$$3x+y=9,2x-3y=6$$

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

$$3x+y=9$$

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

$$3x=-y+9$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(-y+9\right)$$

Whakareatia \frac{1}{3} ki te -y+9.

$$x=-\frac{1}{3}y+3$$

Whakakapia te -\frac{y}{3}+3 mō te x ki tērā atu whārite, 2x-3y=6.

$$2\left(-\frac{1}{3}y+3\right)-3y=6$$

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

$$-\frac{2}{3}y+6-3y=6$$

Tāpiri -\frac{2y}{3} ki te -3y.

$$-\frac{11}{3}y+6=6$$

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

$$-\frac{11}{3}y=0$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{11}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=0$$

Whakaurua te 0 mō y ki x=-\frac{1}{3}y+3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=0$$

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.

$$3x+y=9,2x-3y=6$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&1\\2&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}9\\6\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}3&1\\2&-3\end{matrix}\right))\left(\begin{matrix}3&1\\2&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&1\\2&-3\end{matrix}\right))\left(\begin{matrix}9\\6\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&1\\2&-3\end{matrix}\right))\left(\begin{matrix}9\\6\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&1\\2&-3\end{matrix}\right))\left(\begin{matrix}9\\6\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{3\left(-3\right)-2}&-\frac{1}{3\left(-3\right)-2}\\-\frac{2}{3\left(-3\right)-2}&\frac{3}{3\left(-3\right)-2}\end{matrix}\right)\left(\begin{matrix}9\\6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{11}&\frac{1}{11}\\\frac{2}{11}&-\frac{3}{11}\end{matrix}\right)\left(\begin{matrix}9\\6\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{11}\times 9+\frac{1}{11}\times 6\\\frac{2}{11}\times 9-\frac{3}{11}\times 6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3\\0\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=3,y=0$$

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.

$$3x+y=9,2x-3y=6$$

Kia ōrite ai a 3x 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 3.

$$2\times 3x+2y=2\times 9,3\times 2x+3\left(-3\right)y=3\times 6$$

Whakarūnātia.

$$6x+2y=18,6x-9y=18$$

Me tango 6x-9y=18 mai i 6x+2y=18 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x+2y+9y=18-18$$

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.

$$2y+9y=18-18$$

Tāpiri 2y ki te 9y.

$$11y=18-18$$

Tāpiri 18 ki te -18.

$$11y=0$$

Whakawehea ngā taha e rua ki te 11.

$$y=0$$

Whakaurua te 0 mō y ki 2x-3y=6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x=6$$

Whakawehea ngā taha e rua ki te 2.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=0$$