Whakaoti i te {c}{x+y=12}{x=3y}

$$\left.\begin{array} { c } { x + y = 12 } \\ { x = 3 y } \end{array} \right\}$$

Whakaoti mō x, y

$$x=9$$

$$y=3$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Tangohia te 3y mai i ngā taha e rua.

$$x-3y=0$$

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.

$$x+y=12,x-3y=0$$

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

$$x+y=12$$

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

$$x=-y+12$$

Whakakapia te -y+12 mō te x ki tērā atu whārite, x-3y=0.

$$-y+12-3y=0$$

Tāpiri -y ki te -3y.

$$-4y+12=0$$

Me tango 12 mai i ngā taha e rua o te whārite.

$$-4y=-12$$

Whakawehea ngā taha e rua ki te -4.

$$y=3$$

Whakaurua te 3 mō y ki x=-y+12. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-3+12$$

Tāpiri 12 ki te -3.

$$x=9$$

Kua oti te pūnaha te whakatau.

$$x=9,y=3$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Tangohia te 3y mai i ngā taha e rua.

$$x-3y=0$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$x+y=12,x-3y=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\1&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}12\\0\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&1\\1&-3\end{matrix}\right))\left(\begin{matrix}1&1\\1&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\1&-3\end{matrix}\right))\left(\begin{matrix}12\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\1&-3\end{matrix}\right))\left(\begin{matrix}12\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\1&-3\end{matrix}\right))\left(\begin{matrix}12\\0\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{-3-1}&-\frac{1}{-3-1}\\-\frac{1}{-3-1}&\frac{1}{-3-1}\end{matrix}\right)\left(\begin{matrix}12\\0\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{4}&\frac{1}{4}\\\frac{1}{4}&-\frac{1}{4}\end{matrix}\right)\left(\begin{matrix}12\\0\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{4}\times 12\\\frac{1}{4}\times 12\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}9\\3\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=9,y=3$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Tangohia te 3y mai i ngā taha e rua.

$$x-3y=0$$

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.

$$x+y=12,x-3y=0$$

Me tango x-3y=0 mai i x+y=12 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$x-x+y+3y=12$$

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.

$$y+3y=12$$

Tāpiri y ki te 3y.

$$4y=12$$

Whakawehea ngā taha e rua ki te 4.

$$y=3$$

Whakaurua te 3 mō y ki x-3y=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x-3\times 3=0$$

Whakareatia -3 ki te 3.

$$x-9=0$$

Me tāpiri 9 ki ngā taha e rua o te whārite.

$$x=9$$

Kua oti te pūnaha te whakatau.

$$x=9,y=3$$