Whakaoti i te {l}{2x+y=9}{2x+3y=2}

$$\left. \begin{array} { l } { 2 x + y = 9 } \\ { 2 x + 3 y = 2 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{25}{4} = 6\frac{1}{4} = 6.25$$

$$y = -\frac{7}{2} = -3\frac{1}{2} = -3.5$$

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.

$$2x+y=9,2x+3y=2$$

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

$$2x+y=9$$

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

$$2x=-y+9$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(-y+9\right)$$

Whakareatia \frac{1}{2} ki te -y+9.

$$x=-\frac{1}{2}y+\frac{9}{2}$$

Whakakapia te \frac{-y+9}{2} mō te x ki tērā atu whārite, 2x+3y=2.

$$2\left(-\frac{1}{2}y+\frac{9}{2}\right)+3y=2$$

Whakareatia 2 ki te \frac{-y+9}{2}.

$$-y+9+3y=2$$

Tāpiri -y ki te 3y.

$$2y+9=2$$

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

$$2y=-7$$

Whakawehea ngā taha e rua ki te 2.

$$y=-\frac{7}{2}$$

Whakaurua te -\frac{7}{2} mō y ki x=-\frac{1}{2}y+\frac{9}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{2}\left(-\frac{7}{2}\right)+\frac{9}{2}$$

Whakareatia -\frac{1}{2} ki te -\frac{7}{2} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=\frac{7}{4}+\frac{9}{2}$$

Tāpiri \frac{9}{2} ki te \frac{7}{4} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=\frac{25}{4}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{25}{4},y=-\frac{7}{2}$$

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.

$$2x+y=9,2x+3y=2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&1\\2&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}9\\2\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&1\\2&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&1\\2&3\end{matrix}\right))\left(\begin{matrix}2&1\\2&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\2&3\end{matrix}\right))\left(\begin{matrix}9\\2\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\2&3\end{matrix}\right))\left(\begin{matrix}9\\2\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\2&3\end{matrix}\right))\left(\begin{matrix}9\\2\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{2\times 3-2}&-\frac{1}{2\times 3-2}\\-\frac{2}{2\times 3-2}&\frac{2}{2\times 3-2}\end{matrix}\right)\left(\begin{matrix}9\\2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{4}&-\frac{1}{4}\\-\frac{1}{2}&\frac{1}{2}\end{matrix}\right)\left(\begin{matrix}9\\2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{4}\times 9-\frac{1}{4}\times 2\\-\frac{1}{2}\times 9+\frac{1}{2}\times 2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{25}{4}\\-\frac{7}{2}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{25}{4},y=-\frac{7}{2}$$

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.

$$2x+y=9,2x+3y=2$$

Me tango 2x+3y=2 mai i 2x+y=9 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2x-2x+y-3y=9-2$$

Tāpiri 2x ki te -2x. Ka whakakore atu ngā kupu 2x me -2x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$y-3y=9-2$$

Tāpiri y ki te -3y.

$$-2y=9-2$$

Tāpiri 9 ki te -2.

$$-2y=7$$

Whakawehea ngā taha e rua ki te -2.

$$y=-\frac{7}{2}$$

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

$$2x+3\left(-\frac{7}{2}\right)=2$$

Whakareatia 3 ki te -\frac{7}{2}.

$$2x-\frac{21}{2}=2$$

Me tāpiri \frac{21}{2} ki ngā taha e rua o te whārite.

$$2x=\frac{25}{2}$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{25}{4}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{25}{4},y=-\frac{7}{2}$$