Whakaoti i te {l}{3x-2y=0}{2x+3y=9}

$$\left\{ \begin{array} { l } { 3 x - 2 y = 0 } \\ { 2 x + 3 y = 9 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{18}{13} = 1\frac{5}{13} \approx 1.384615385$$

$$y = \frac{27}{13} = 2\frac{1}{13} \approx 2.076923077$$

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.

$$3x-2y=0,2x+3y=9$$

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

$$3x-2y=0$$

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

$$3x=2y$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\times 2y$$

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

$$x=\frac{2}{3}y$$

Whakakapia te \frac{2y}{3} mō te x ki tērā atu whārite, 2x+3y=9.

$$2\times \frac{2}{3}y+3y=9$$

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

$$\frac{4}{3}y+3y=9$$

Tāpiri \frac{4y}{3} ki te 3y.

$$\frac{13}{3}y=9$$

Whakawehea ngā taha e rua o te whārite ki te \frac{13}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{27}{13}$$

Whakaurua te \frac{27}{13} mō y ki x=\frac{2}{3}y. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{2}{3}\times \frac{27}{13}$$

Whakareatia \frac{2}{3} ki te \frac{27}{13} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=\frac{18}{13}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{18}{13},y=\frac{27}{13}$$

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.

$$3x-2y=0,2x+3y=9$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{13}&\frac{2}{13}\\-\frac{2}{13}&\frac{3}{13}\end{matrix}\right)\left(\begin{matrix}0\\9\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{13}\times 9\\\frac{3}{13}\times 9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{18}{13}\\\frac{27}{13}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{18}{13},y=\frac{27}{13}$$

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.

$$3x-2y=0,2x+3y=9$$

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.

$$2\times 3x+2\left(-2\right)y=0,3\times 2x+3\times 3y=3\times 9$$

Whakarūnātia.

$$6x-4y=0,6x+9y=27$$

Me tango 6x+9y=27 mai i 6x-4y=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x-4y-9y=-27$$

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.

$$-4y-9y=-27$$

Tāpiri -4y ki te -9y.

$$-13y=-27$$

Whakawehea ngā taha e rua ki te -13.

$$y=\frac{27}{13}$$

Whakaurua te \frac{27}{13} mō y ki 2x+3y=9. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x+3\times \frac{27}{13}=9$$

Whakareatia 3 ki te \frac{27}{13}.

$$2x+\frac{81}{13}=9$$

Me tango \frac{81}{13} mai i ngā taha e rua o te whārite.

$$2x=\frac{36}{13}$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{18}{13}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{18}{13},y=\frac{27}{13}$$