Whakaoti i te {l}{x+y=1}{x-2y=14}

$$\left. \begin{array} { l } { x + y = 1 } \\ { x - 2 y = 14 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{16}{3} = 5\frac{1}{3} \approx 5.333333333$$

$$y = -\frac{13}{3} = -4\frac{1}{3} \approx -4.333333333$$

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.

$$x+y=1,x-2y=14$$

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

$$x+y=1$$

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

$$x=-y+1$$

Whakakapia te -y+1 mō te x ki tērā atu whārite, x-2y=14.

$$-y+1-2y=14$$

Tāpiri -y ki te -2y.

$$-3y+1=14$$

Me tango 1 mai i ngā taha e rua o te whārite.

$$-3y=13$$

Whakawehea ngā taha e rua ki te -3.

$$y=-\frac{13}{3}$$

Whakaurua te -\frac{13}{3} mō y ki x=-y+1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\left(-\frac{13}{3}\right)+1$$

Whakareatia -1 ki te -\frac{13}{3}.

$$x=\frac{13}{3}+1$$

Tāpiri 1 ki te \frac{13}{3}.

$$x=\frac{16}{3}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{16}{3},y=-\frac{13}{3}$$

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.

$$x+y=1,x-2y=14$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\1&-2\end{matrix}\right))\left(\begin{matrix}1\\14\end{matrix}\right)$$

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{16}{3}\\-\frac{13}{3}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{16}{3},y=-\frac{13}{3}$$

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.

$$x+y=1,x-2y=14$$

Me tango x-2y=14 mai i x+y=1 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$x-x+y+2y=1-14$$

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.

$$y+2y=1-14$$

Tāpiri y ki te 2y.

$$3y=1-14$$

Tāpiri 1 ki te -14.

$$3y=-13$$

Whakawehea ngā taha e rua ki te 3.

$$y=-\frac{13}{3}$$

Whakaurua te -\frac{13}{3} mō y ki x-2y=14. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x-2\left(-\frac{13}{3}\right)=14$$

Whakareatia -2 ki te -\frac{13}{3}.

$$x+\frac{26}{3}=14$$

Me tango \frac{26}{3} mai i ngā taha e rua o te whārite.

$$x=\frac{16}{3}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{16}{3},y=-\frac{13}{3}$$