Whakaoti i te {l}{3x+2y=32}{-x+3y=15}

$$\left. \begin{array} { l } { 3 x + 2 y = 32 } \\ { - x + 3 y = 15 } \end{array} \right.$$

Whakaoti mō x, y

$$x=6$$

$$y=7$$

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.

$$3x+2y=32,-x+3y=15$$

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

$$3x+2y=32$$

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

$$3x=-2y+32$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(-2y+32\right)$$

Whakareatia \frac{1}{3} ki te -2y+32.

$$x=-\frac{2}{3}y+\frac{32}{3}$$

Whakakapia te \frac{-2y+32}{3} mō te x ki tērā atu whārite, -x+3y=15.

$$-\left(-\frac{2}{3}y+\frac{32}{3}\right)+3y=15$$

Whakareatia -1 ki te \frac{-2y+32}{3}.

$$\frac{2}{3}y-\frac{32}{3}+3y=15$$

Tāpiri \frac{2y}{3} ki te 3y.

$$\frac{11}{3}y-\frac{32}{3}=15$$

Me tāpiri \frac{32}{3} ki ngā taha e rua o te whārite.

$$\frac{11}{3}y=\frac{77}{3}$$

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.

$$y=7$$

Whakaurua te 7 mō y ki x=-\frac{2}{3}y+\frac{32}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{2}{3}\times 7+\frac{32}{3}$$

Whakareatia -\frac{2}{3} ki te 7.

$$x=\frac{-14+32}{3}$$

Tāpiri \frac{32}{3} ki te -\frac{14}{3} 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.

$$x=6$$

Kua oti te pūnaha te whakatau.

$$x=6,y=7$$

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.

$$3x+2y=32,-x+3y=15$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&2\\-1&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}32\\15\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}3&2\\-1&3\end{matrix}\right))\left(\begin{matrix}3&2\\-1&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&2\\-1&3\end{matrix}\right))\left(\begin{matrix}32\\15\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&2\\-1&3\end{matrix}\right))\left(\begin{matrix}32\\15\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&2\\-1&3\end{matrix}\right))\left(\begin{matrix}32\\15\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{3\times 3-2\left(-1\right)}&-\frac{2}{3\times 3-2\left(-1\right)}\\-\frac{-1}{3\times 3-2\left(-1\right)}&\frac{3}{3\times 3-2\left(-1\right)}\end{matrix}\right)\left(\begin{matrix}32\\15\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{11}&-\frac{2}{11}\\\frac{1}{11}&\frac{3}{11}\end{matrix}\right)\left(\begin{matrix}32\\15\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{11}\times 32-\frac{2}{11}\times 15\\\frac{1}{11}\times 32+\frac{3}{11}\times 15\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}6\\7\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=6,y=7$$

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.

$$3x+2y=32,-x+3y=15$$

Kia ōrite ai a 3x me -x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -1 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$-3x-2y=-32,3\left(-1\right)x+3\times 3y=3\times 15$$

Whakarūnātia.

$$-3x-2y=-32,-3x+9y=45$$

Me tango -3x+9y=45 mai i -3x-2y=-32 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-3x+3x-2y-9y=-32-45$$

Tāpiri -3x ki te 3x. Ka whakakore atu ngā kupu -3x me 3x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-2y-9y=-32-45$$

Tāpiri -2y ki te -9y.

$$-11y=-32-45$$

Tāpiri -32 ki te -45.

$$-11y=-77$$

Whakawehea ngā taha e rua ki te -11.

$$y=7$$

Whakaurua te 7 mō y ki -x+3y=15. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-x+3\times 7=15$$

Whakareatia 3 ki te 7.

$$-x+21=15$$

Me tango 21 mai i ngā taha e rua o te whārite.

$$-x=-6$$

Whakawehea ngā taha e rua ki te -1.

$$x=6$$

Kua oti te pūnaha te whakatau.

$$x=6,y=7$$