Whakaoti i te {l}{x+y=60}{x-2y=0}

$$\left\{ \begin{array} { l } { x + y = 60 } \\ { x - 2 y = 0 } \end{array} \right.$$

Whakaoti mō x, y

$$x=40$$

$$y=20$$

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.

$$x+y=60,x-2y=0$$

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

$$x+y=60$$

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

$$x=-y+60$$

Whakakapia te -y+60 mō te x ki tērā atu whārite, x-2y=0.

$$-y+60-2y=0$$

Tāpiri -y ki te -2y.

$$-3y+60=0$$

Me tango 60 mai i ngā taha e rua o te whārite.

$$-3y=-60$$

Whakawehea ngā taha e rua ki te -3.

$$y=20$$

Whakaurua te 20 mō y ki x=-y+60. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-20+60$$

Tāpiri 60 ki te -20.

$$x=40$$

Kua oti te pūnaha te whakatau.

$$x=40,y=20$$

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.

$$x+y=60,x-2y=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\1&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}60\\0\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&1\\1&-2\end{matrix}\right))\left(\begin{matrix}1&1\\1&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\1&-2\end{matrix}\right))\left(\begin{matrix}60\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\1&-2\end{matrix}\right))\left(\begin{matrix}60\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\1&-2\end{matrix}\right))\left(\begin{matrix}60\\0\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{-2-1}&-\frac{1}{-2-1}\\-\frac{1}{-2-1}&\frac{1}{-2-1}\end{matrix}\right)\left(\begin{matrix}60\\0\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{3}&\frac{1}{3}\\\frac{1}{3}&-\frac{1}{3}\end{matrix}\right)\left(\begin{matrix}60\\0\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{3}\times 60\\\frac{1}{3}\times 60\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}40\\20\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=40,y=20$$

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.

$$x+y=60,x-2y=0$$

Me tango x-2y=0 mai i x+y=60 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$x-x+y+2y=60$$

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.

$$y+2y=60$$

Tāpiri y ki te 2y.

$$3y=60$$

Whakawehea ngā taha e rua ki te 3.

$$y=20$$

Whakaurua te 20 mō y ki x-2y=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x-2\times 20=0$$

Whakareatia -2 ki te 20.

$$x-40=0$$

Me tāpiri 40 ki ngā taha e rua o te whārite.

$$x=40$$

Kua oti te pūnaha te whakatau.

$$x=40,y=20$$