Whakaoti i te {l}{3x+2y=1}{2x-7y=-2}

$$\left\{ \begin{array} { l } { 3 x + 2 y = 1 } \\ { 2 x - 7 y = - 2 } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{3}{25}=0.12$$

$$y=\frac{8}{25}=0.32$$

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.

$$3x+2y=1,2x-7y=-2$$

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

$$3x+2y=1$$

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

$$3x=-2y+1$$

Whakawehea ngā taha e rua ki te 3.

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

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

$$x=-\frac{2}{3}y+\frac{1}{3}$$

Whakakapia te \frac{-2y+1}{3} mō te x ki tērā atu whārite, 2x-7y=-2.

$$2\left(-\frac{2}{3}y+\frac{1}{3}\right)-7y=-2$$

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

$$-\frac{4}{3}y+\frac{2}{3}-7y=-2$$

Tāpiri -\frac{4y}{3} ki te -7y.

$$-\frac{25}{3}y+\frac{2}{3}=-2$$

Me tango \frac{2}{3} mai i ngā taha e rua o te whārite.

$$-\frac{25}{3}y=-\frac{8}{3}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{25}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{8}{25}$$

Whakaurua te \frac{8}{25} mō y ki x=-\frac{2}{3}y+\frac{1}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{2}{3}\times \frac{8}{25}+\frac{1}{3}$$

Whakareatia -\frac{2}{3} ki te \frac{8}{25} 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.

$$x=-\frac{16}{75}+\frac{1}{3}$$

Tāpiri \frac{1}{3} ki te -\frac{16}{75} 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.

$$x=\frac{3}{25}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{3}{25},y=\frac{8}{25}$$

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.

$$3x+2y=1,2x-7y=-2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&2\\2&-7\end{matrix}\right).

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

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&2\\2&-7\end{matrix}\right))\left(\begin{matrix}1\\-2\end{matrix}\right)$$

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{25}&\frac{2}{25}\\\frac{2}{25}&-\frac{3}{25}\end{matrix}\right)\left(\begin{matrix}1\\-2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{25}+\frac{2}{25}\left(-2\right)\\\frac{2}{25}-\frac{3}{25}\left(-2\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{25}\\\frac{8}{25}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{3}{25},y=\frac{8}{25}$$

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.

$$3x+2y=1,2x-7y=-2$$

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.

$$2\times 3x+2\times 2y=2,3\times 2x+3\left(-7\right)y=3\left(-2\right)$$

Whakarūnātia.

$$6x+4y=2,6x-21y=-6$$

Me tango 6x-21y=-6 mai i 6x+4y=2 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x+4y+21y=2+6$$

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.

$$4y+21y=2+6$$

Tāpiri 4y ki te 21y.

$$25y=2+6$$

Tāpiri 2 ki te 6.

$$25y=8$$

Whakawehea ngā taha e rua ki te 25.

$$y=\frac{8}{25}$$

Whakaurua te \frac{8}{25} mō y ki 2x-7y=-2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x-7\times \frac{8}{25}=-2$$

Whakareatia -7 ki te \frac{8}{25}.

$$2x-\frac{56}{25}=-2$$

Me tāpiri \frac{56}{25} ki ngā taha e rua o te whārite.

$$2x=\frac{6}{25}$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{3}{25}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{3}{25},y=\frac{8}{25}$$