Whakaoti i te {l}{7x-2y=11}{x+y=8}

$$\left. \begin{array} { l } { 7 x - 2 y = 11 } \\ { x + y = 8 } \end{array} \right.$$

Whakaoti mō x, y

$$x=3$$

$$y=5$$

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.

$$7x-2y=11,x+y=8$$

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

$$7x-2y=11$$

Me tāpiri 2y ki ngā taha e rua o te whārite.

$$7x=2y+11$$

Whakawehea ngā taha e rua ki te 7.

$$x=\frac{1}{7}\left(2y+11\right)$$

Whakareatia \frac{1}{7} ki te 2y+11.

$$x=\frac{2}{7}y+\frac{11}{7}$$

Whakakapia te \frac{2y+11}{7} mō te x ki tērā atu whārite, x+y=8.

$$\frac{2}{7}y+\frac{11}{7}+y=8$$

Tāpiri \frac{2y}{7} ki te y.

$$\frac{9}{7}y+\frac{11}{7}=8$$

Me tango \frac{11}{7} mai i ngā taha e rua o te whārite.

$$\frac{9}{7}y=\frac{45}{7}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{9}{7}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=5$$

Whakaurua te 5 mō y ki x=\frac{2}{7}y+\frac{11}{7}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{2}{7}\times 5+\frac{11}{7}$$

Whakareatia \frac{2}{7} ki te 5.

$$x=\frac{10+11}{7}$$

Tāpiri \frac{11}{7} ki te \frac{10}{7} 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.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=5$$

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.

$$7x-2y=11,x+y=8$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}7&-2\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}11\\8\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}7&-2\\1&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}7&-2\\1&1\end{matrix}\right))\left(\begin{matrix}7&-2\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&-2\\1&1\end{matrix}\right))\left(\begin{matrix}11\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&-2\\1&1\end{matrix}\right))\left(\begin{matrix}11\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&-2\\1&1\end{matrix}\right))\left(\begin{matrix}11\\8\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{7-\left(-2\right)}&-\frac{-2}{7-\left(-2\right)}\\-\frac{1}{7-\left(-2\right)}&\frac{7}{7-\left(-2\right)}\end{matrix}\right)\left(\begin{matrix}11\\8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{9}&\frac{2}{9}\\-\frac{1}{9}&\frac{7}{9}\end{matrix}\right)\left(\begin{matrix}11\\8\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{9}\times 11+\frac{2}{9}\times 8\\-\frac{1}{9}\times 11+\frac{7}{9}\times 8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3\\5\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=3,y=5$$

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.

$$7x-2y=11,x+y=8$$

Kia ōrite ai a 7x 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 7.

$$7x-2y=11,7x+7y=7\times 8$$

Whakarūnātia.

$$7x-2y=11,7x+7y=56$$

Me tango 7x+7y=56 mai i 7x-2y=11 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$7x-7x-2y-7y=11-56$$

Tāpiri 7x ki te -7x. Ka whakakore atu ngā kupu 7x me -7x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-2y-7y=11-56$$

Tāpiri -2y ki te -7y.

$$-9y=11-56$$

Tāpiri 11 ki te -56.

$$-9y=-45$$

Whakawehea ngā taha e rua ki te -9.

$$y=5$$

Whakaurua te 5 mō y ki x+y=8. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x+5=8$$

Me tango 5 mai i ngā taha e rua o te whārite.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=5$$