Whakaoti i te {l}{x+2y=7}{y-x=1}

$$\left\{ \begin{array} { l } { x + 2 y = 7 } \\ { y - x = 1 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{5}{3} = 1\frac{2}{3} \approx 1.666666667$$

$$y = \frac{8}{3} = 2\frac{2}{3} \approx 2.666666667$$

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.

$$x+2y=7,-x+y=1$$

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

$$x+2y=7$$

Me tango 2y mai i ngā taha e rua o te whārite.

$$x=-2y+7$$

Whakakapia te -2y+7 mō te x ki tērā atu whārite, -x+y=1.

$$-\left(-2y+7\right)+y=1$$

Whakareatia -1 ki te -2y+7.

$$2y-7+y=1$$

Tāpiri 2y ki te y.

$$3y-7=1$$

Me tāpiri 7 ki ngā taha e rua o te whārite.

$$3y=8$$

Whakawehea ngā taha e rua ki te 3.

$$y=\frac{8}{3}$$

Whakaurua te \frac{8}{3} mō y ki x=-2y+7. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-2\times \frac{8}{3}+7$$

Whakareatia -2 ki te \frac{8}{3}.

$$x=-\frac{16}{3}+7$$

Tāpiri 7 ki te -\frac{16}{3}.

$$x=\frac{5}{3}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{5}{3},y=\frac{8}{3}$$

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.

$$x+2y=7,-x+y=1$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&2\\-1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}7\\1\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&2\\-1&1\end{matrix}\right))\left(\begin{matrix}1&2\\-1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\-1&1\end{matrix}\right))\left(\begin{matrix}7\\1\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\-1&1\end{matrix}\right))\left(\begin{matrix}7\\1\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\-1&1\end{matrix}\right))\left(\begin{matrix}7\\1\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{1-2\left(-1\right)}&-\frac{2}{1-2\left(-1\right)}\\-\frac{-1}{1-2\left(-1\right)}&\frac{1}{1-2\left(-1\right)}\end{matrix}\right)\left(\begin{matrix}7\\1\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3}&-\frac{2}{3}\\\frac{1}{3}&\frac{1}{3}\end{matrix}\right)\left(\begin{matrix}7\\1\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3}\times 7-\frac{2}{3}\\\frac{1}{3}\times 7+\frac{1}{3}\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{3}\\\frac{8}{3}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{5}{3},y=\frac{8}{3}$$

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.

$$x+2y=7,-x+y=1$$

Kia ōrite ai a x 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 1.

$$-x-2y=-7,-x+y=1$$

Me tango -x+y=1 mai i -x-2y=-7 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-x+x-2y-y=-7-1$$

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.

$$-2y-y=-7-1$$

Tāpiri -2y ki te -y.

$$-3y=-7-1$$

Tāpiri -7 ki te -1.

$$-3y=-8$$

Whakawehea ngā taha e rua ki te -3.

$$y=\frac{8}{3}$$

Whakaurua te \frac{8}{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.

$$-x+\frac{8}{3}=1$$

Me tango \frac{8}{3} mai i ngā taha e rua o te whārite.

$$-x=-\frac{5}{3}$$

Whakawehea ngā taha e rua ki te -1.

$$x=\frac{5}{3}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{5}{3},y=\frac{8}{3}$$