Whakaoti i te {l}{x+2y=16}{2x+3y=17}

$$\left. \begin{array} { l } { x + 2 y = 16 } \\ { 2 x + 3 y = 17 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-14$$

$$y=15$$

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.

$$x+2y=16,2x+3y=17$$

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

$$x+2y=16$$

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

$$x=-2y+16$$

Whakakapia te -2y+16 mō te x ki tērā atu whārite, 2x+3y=17.

$$2\left(-2y+16\right)+3y=17$$

Whakareatia 2 ki te -2y+16.

$$-4y+32+3y=17$$

Tāpiri -4y ki te 3y.

$$-y+32=17$$

Me tango 32 mai i ngā taha e rua o te whārite.

$$-y=-15$$

Whakawehea ngā taha e rua ki te -1.

$$y=15$$

Whakaurua te 15 mō y ki x=-2y+16. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-2\times 15+16$$

Whakareatia -2 ki te 15.

$$x=-30+16$$

Tāpiri 16 ki te -30.

$$x=-14$$

Kua oti te pūnaha te whakatau.

$$x=-14,y=15$$

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.

$$x+2y=16,2x+3y=17$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&2\\2&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}16\\17\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&2\\2&3\end{matrix}\right))\left(\begin{matrix}1&2\\2&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\2&3\end{matrix}\right))\left(\begin{matrix}16\\17\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\2&3\end{matrix}\right))\left(\begin{matrix}16\\17\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\2&3\end{matrix}\right))\left(\begin{matrix}16\\17\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{3-2\times 2}&-\frac{2}{3-2\times 2}\\-\frac{2}{3-2\times 2}&\frac{1}{3-2\times 2}\end{matrix}\right)\left(\begin{matrix}16\\17\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-3&2\\2&-1\end{matrix}\right)\left(\begin{matrix}16\\17\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-3\times 16+2\times 17\\2\times 16-17\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-14\\15\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-14,y=15$$

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.

$$x+2y=16,2x+3y=17$$

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.

$$2x+2\times 2y=2\times 16,2x+3y=17$$

Whakarūnātia.

$$2x+4y=32,2x+3y=17$$

Me tango 2x+3y=17 mai i 2x+4y=32 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2x-2x+4y-3y=32-17$$

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.

$$4y-3y=32-17$$

Tāpiri 4y ki te -3y.

$$y=32-17$$

Tāpiri 32 ki te -17.

$$y=15$$

Whakaurua te 15 mō y ki 2x+3y=17. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x+3\times 15=17$$

Whakareatia 3 ki te 15.

$$2x+45=17$$

Me tango 45 mai i ngā taha e rua o te whārite.

$$2x=-28$$

Whakawehea ngā taha e rua ki te 2.

$$x=-14$$

Kua oti te pūnaha te whakatau.

$$x=-14,y=15$$