Whakaoti i te {l}{x+2y=10}{3y-2x=5}

$$\left. \begin{array} { l } { x + 2 y = 10 } \\ { 3 y - 2 x = 5 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{20}{7} = 2\frac{6}{7} \approx 2.857142857$$

$$y = \frac{25}{7} = 3\frac{4}{7} \approx 3.571428571$$

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.

$$x+2y=10,-2x+3y=5$$

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

$$x+2y=10$$

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

$$x=-2y+10$$

Whakakapia te -2y+10 mō te x ki tērā atu whārite, -2x+3y=5.

$$-2\left(-2y+10\right)+3y=5$$

Whakareatia -2 ki te -2y+10.

$$4y-20+3y=5$$

Tāpiri 4y ki te 3y.

$$7y-20=5$$

Me tāpiri 20 ki ngā taha e rua o te whārite.

$$7y=25$$

Whakawehea ngā taha e rua ki te 7.

$$y=\frac{25}{7}$$

Whakaurua te \frac{25}{7} mō y ki x=-2y+10. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-2\times \frac{25}{7}+10$$

Whakareatia -2 ki te \frac{25}{7}.

$$x=-\frac{50}{7}+10$$

Tāpiri 10 ki te -\frac{50}{7}.

$$x=\frac{20}{7}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{20}{7},y=\frac{25}{7}$$

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.

$$x+2y=10,-2x+3y=5$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&2\\-2&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}10\\5\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&2\\-2&3\end{matrix}\right))\left(\begin{matrix}1&2\\-2&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\-2&3\end{matrix}\right))\left(\begin{matrix}10\\5\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\-2&3\end{matrix}\right))\left(\begin{matrix}10\\5\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\-2&3\end{matrix}\right))\left(\begin{matrix}10\\5\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{3-2\left(-2\right)}&-\frac{2}{3-2\left(-2\right)}\\-\frac{-2}{3-2\left(-2\right)}&\frac{1}{3-2\left(-2\right)}\end{matrix}\right)\left(\begin{matrix}10\\5\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{7}&-\frac{2}{7}\\\frac{2}{7}&\frac{1}{7}\end{matrix}\right)\left(\begin{matrix}10\\5\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{7}\times 10-\frac{2}{7}\times 5\\\frac{2}{7}\times 10+\frac{1}{7}\times 5\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{20}{7}\\\frac{25}{7}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{20}{7},y=\frac{25}{7}$$

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.

$$x+2y=10,-2x+3y=5$$

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.

$$-2x-2\times 2y=-2\times 10,-2x+3y=5$$

Whakarūnātia.

$$-2x-4y=-20,-2x+3y=5$$

Me tango -2x+3y=5 mai i -2x-4y=-20 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-2x+2x-4y-3y=-20-5$$

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.

$$-4y-3y=-20-5$$

Tāpiri -4y ki te -3y.

$$-7y=-20-5$$

Tāpiri -20 ki te -5.

$$-7y=-25$$

Whakawehea ngā taha e rua ki te -7.

$$y=\frac{25}{7}$$

Whakaurua te \frac{25}{7} mō y ki -2x+3y=5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-2x+3\times \frac{25}{7}=5$$

Whakareatia 3 ki te \frac{25}{7}.

$$-2x+\frac{75}{7}=5$$

Me tango \frac{75}{7} mai i ngā taha e rua o te whārite.

$$-2x=-\frac{40}{7}$$

Whakawehea ngā taha e rua ki te -2.

$$x=\frac{20}{7}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{20}{7},y=\frac{25}{7}$$