Whakaoti i te {l}{11x+3y=14}{x+7y=8}

$$\left. \begin{array} { l } { 11 x + 3 y = 14 } \\ { x + 7 y = 8 } \end{array} \right.$$

Whakaoti mō x, y

$$x=1$$

$$y=1$$

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.

$$11x+3y=14,x+7y=8$$

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

$$11x+3y=14$$

Me tango 3y mai i ngā taha e rua o te whārite.

$$11x=-3y+14$$

Whakawehea ngā taha e rua ki te 11.

$$x=\frac{1}{11}\left(-3y+14\right)$$

Whakareatia \frac{1}{11} ki te -3y+14.

$$x=-\frac{3}{11}y+\frac{14}{11}$$

Whakakapia te \frac{-3y+14}{11} mō te x ki tērā atu whārite, x+7y=8.

$$-\frac{3}{11}y+\frac{14}{11}+7y=8$$

Tāpiri -\frac{3y}{11} ki te 7y.

$$\frac{74}{11}y+\frac{14}{11}=8$$

Me tango \frac{14}{11} mai i ngā taha e rua o te whārite.

$$\frac{74}{11}y=\frac{74}{11}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{74}{11}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=1$$

Whakaurua te 1 mō y ki x=-\frac{3}{11}y+\frac{14}{11}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{-3+14}{11}$$

Tāpiri \frac{14}{11} ki te -\frac{3}{11} 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.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=1$$

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.

$$11x+3y=14,x+7y=8$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}11&3\\1&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}14\\8\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}11&3\\1&7\end{matrix}\right).

$$inverse(\left(\begin{matrix}11&3\\1&7\end{matrix}\right))\left(\begin{matrix}11&3\\1&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}11&3\\1&7\end{matrix}\right))\left(\begin{matrix}14\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}11&3\\1&7\end{matrix}\right))\left(\begin{matrix}14\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}11&3\\1&7\end{matrix}\right))\left(\begin{matrix}14\\8\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{11\times 7-3}&-\frac{3}{11\times 7-3}\\-\frac{1}{11\times 7-3}&\frac{11}{11\times 7-3}\end{matrix}\right)\left(\begin{matrix}14\\8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{74}&-\frac{3}{74}\\-\frac{1}{74}&\frac{11}{74}\end{matrix}\right)\left(\begin{matrix}14\\8\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{74}\times 14-\frac{3}{74}\times 8\\-\frac{1}{74}\times 14+\frac{11}{74}\times 8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1\\1\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=1,y=1$$

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.

$$11x+3y=14,x+7y=8$$

Kia ōrite ai a 11x 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 11.

$$11x+3y=14,11x+11\times 7y=11\times 8$$

Whakarūnātia.

$$11x+3y=14,11x+77y=88$$

Me tango 11x+77y=88 mai i 11x+3y=14 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$11x-11x+3y-77y=14-88$$

Tāpiri 11x ki te -11x. Ka whakakore atu ngā kupu 11x me -11x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$3y-77y=14-88$$

Tāpiri 3y ki te -77y.

$$-74y=14-88$$

Tāpiri 14 ki te -88.

$$-74y=-74$$

Whakawehea ngā taha e rua ki te -74.

$$y=1$$

Whakaurua te 1 mō y ki x+7y=8. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x+7=8$$

Me tango 7 mai i ngā taha e rua o te whārite.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=1$$