Whakaoti i te {l}{x+2y=11}{x-5y=-17}

$$\left. \begin{array} { l } { x + 2 y = 11 } \\ { x - 5 y = - 17 } \end{array} \right.$$

Whakaoti mō x, y

$$x=3$$

$$y=4$$

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.

$$x+2y=11,x-5y=-17$$

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

$$x+2y=11$$

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

$$x=-2y+11$$

Whakakapia te -2y+11 mō te x ki tērā atu whārite, x-5y=-17.

$$-2y+11-5y=-17$$

Tāpiri -2y ki te -5y.

$$-7y+11=-17$$

Me tango 11 mai i ngā taha e rua o te whārite.

$$-7y=-28$$

Whakawehea ngā taha e rua ki te -7.

$$y=4$$

Whakaurua te 4 mō y ki x=-2y+11. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-2\times 4+11$$

Whakareatia -2 ki te 4.

$$x=-8+11$$

Tāpiri 11 ki te -8.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=4$$

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.

$$x+2y=11,x-5y=-17$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&2\\1&-5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}11\\-17\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&2\\1&-5\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&2\\1&-5\end{matrix}\right))\left(\begin{matrix}1&2\\1&-5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\1&-5\end{matrix}\right))\left(\begin{matrix}11\\-17\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\1&-5\end{matrix}\right))\left(\begin{matrix}11\\-17\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\1&-5\end{matrix}\right))\left(\begin{matrix}11\\-17\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{-5-2}&-\frac{2}{-5-2}\\-\frac{1}{-5-2}&\frac{1}{-5-2}\end{matrix}\right)\left(\begin{matrix}11\\-17\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{7}&\frac{2}{7}\\\frac{1}{7}&-\frac{1}{7}\end{matrix}\right)\left(\begin{matrix}11\\-17\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{7}\times 11+\frac{2}{7}\left(-17\right)\\\frac{1}{7}\times 11-\frac{1}{7}\left(-17\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3\\4\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=3,y=4$$

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.

$$x+2y=11,x-5y=-17$$

Me tango x-5y=-17 mai i x+2y=11 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$x-x+2y+5y=11+17$$

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

$$2y+5y=11+17$$

Tāpiri 2y ki te 5y.

$$7y=11+17$$

Tāpiri 11 ki te 17.

$$7y=28$$

Whakawehea ngā taha e rua ki te 7.

$$y=4$$

Whakaurua te 4 mō y ki x-5y=-17. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x-5\times 4=-17$$

Whakareatia -5 ki te 4.

$$x-20=-17$$

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

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=4$$