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

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

Whakaoti mō x, y

$$x=5$$

$$y=3$$

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.

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

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

$$x+y=8$$

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

$$x=-y+8$$

Whakakapia te -y+8 mō te x ki tērā atu whārite, x+3y=14.

$$-y+8+3y=14$$

Tāpiri -y ki te 3y.

$$2y+8=14$$

Me tango 8 mai i ngā taha e rua o te whārite.

$$2y=6$$

Whakawehea ngā taha e rua ki te 2.

$$y=3$$

Whakaurua te 3 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.

$$x=-3+8$$

Tāpiri 8 ki te -3.

$$x=5$$

Kua oti te pūnaha te whakatau.

$$x=5,y=3$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te poukapa kōaro ko \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), kia tuhia anō ai te whārite poukapa hei rapanga whakarea poukapa.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{2}&-\frac{1}{2}\\-\frac{1}{2}&\frac{1}{2}\end{matrix}\right)\left(\begin{matrix}8\\14\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{2}\times 8-\frac{1}{2}\times 14\\-\frac{1}{2}\times 8+\frac{1}{2}\times 14\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=5,y=3$$

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.

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

Me tango x+3y=14 mai i x+y=8 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$x-x+y-3y=8-14$$

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.

$$y-3y=8-14$$

Tāpiri y ki te -3y.

$$-2y=8-14$$

Tāpiri 8 ki te -14.

$$-2y=-6$$

Whakawehea ngā taha e rua ki te -2.

$$y=3$$

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

$$x+3\times 3=14$$

Whakareatia 3 ki te 3.

$$x+9=14$$

Me tango 9 mai i ngā taha e rua o te whārite.

$$x=5$$

Kua oti te pūnaha te whakatau.

$$x=5,y=3$$