Whakaoti i te {l}{3x+2y=8}{2x+3y=9}

$$\left. \begin{array} { l } { 3 x + 2 y = 8 } \\ { 2 x + 3 y = 9 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{6}{5} = 1\frac{1}{5} = 1.2$$

$$y = \frac{11}{5} = 2\frac{1}{5} = 2.2$$

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.

$$3x+2y=8,2x+3y=9$$

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

$$3x+2y=8$$

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

$$3x=-2y+8$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(-2y+8\right)$$

Whakareatia \frac{1}{3} ki te -2y+8.

$$x=-\frac{2}{3}y+\frac{8}{3}$$

Whakakapia te \frac{-2y+8}{3} mō te x ki tērā atu whārite, 2x+3y=9.

$$2\left(-\frac{2}{3}y+\frac{8}{3}\right)+3y=9$$

Whakareatia 2 ki te \frac{-2y+8}{3}.

$$-\frac{4}{3}y+\frac{16}{3}+3y=9$$

Tāpiri -\frac{4y}{3} ki te 3y.

$$\frac{5}{3}y+\frac{16}{3}=9$$

Me tango \frac{16}{3} mai i ngā taha e rua o te whārite.

$$\frac{5}{3}y=\frac{11}{3}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{5}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{11}{5}$$

Whakaurua te \frac{11}{5} mō y ki x=-\frac{2}{3}y+\frac{8}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{2}{3}\times \frac{11}{5}+\frac{8}{3}$$

Whakareatia -\frac{2}{3} ki te \frac{11}{5} 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.

$$x=-\frac{22}{15}+\frac{8}{3}$$

Tāpiri \frac{8}{3} ki te -\frac{22}{15} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=\frac{6}{5}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{6}{5},y=\frac{11}{5}$$

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.

$$3x+2y=8,2x+3y=9$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&2\\2&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}8\\9\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}3&2\\2&3\end{matrix}\right))\left(\begin{matrix}3&2\\2&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&2\\2&3\end{matrix}\right))\left(\begin{matrix}8\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&2\\2&3\end{matrix}\right))\left(\begin{matrix}8\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&2\\2&3\end{matrix}\right))\left(\begin{matrix}8\\9\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{3\times 3-2\times 2}&-\frac{2}{3\times 3-2\times 2}\\-\frac{2}{3\times 3-2\times 2}&\frac{3}{3\times 3-2\times 2}\end{matrix}\right)\left(\begin{matrix}8\\9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{5}&-\frac{2}{5}\\-\frac{2}{5}&\frac{3}{5}\end{matrix}\right)\left(\begin{matrix}8\\9\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{5}\times 8-\frac{2}{5}\times 9\\-\frac{2}{5}\times 8+\frac{3}{5}\times 9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{6}{5}\\\frac{11}{5}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{6}{5},y=\frac{11}{5}$$

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.

$$3x+2y=8,2x+3y=9$$

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.

$$2\times 3x+2\times 2y=2\times 8,3\times 2x+3\times 3y=3\times 9$$

Whakarūnātia.

$$6x+4y=16,6x+9y=27$$

Me tango 6x+9y=27 mai i 6x+4y=16 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x+4y-9y=16-27$$

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.

$$4y-9y=16-27$$

Tāpiri 4y ki te -9y.

$$-5y=16-27$$

Tāpiri 16 ki te -27.

$$-5y=-11$$

Whakawehea ngā taha e rua ki te -5.

$$y=\frac{11}{5}$$

Whakaurua te \frac{11}{5} 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.

$$2x+3\times \frac{11}{5}=9$$

Whakareatia 3 ki te \frac{11}{5}.

$$2x+\frac{33}{5}=9$$

Me tango \frac{33}{5} mai i ngā taha e rua o te whārite.

$$2x=\frac{12}{5}$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{6}{5}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{6}{5},y=\frac{11}{5}$$