Whakaoti i te x+2y=12,12=9x-2y

$$x + 2 y = 12,12 = 9 x - 2 y$$

Whakaoti mō x, y

$$x = \frac{12}{5} = 2\frac{2}{5} = 2.4$$

$$y = \frac{24}{5} = 4\frac{4}{5} = 4.8$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$9x-2y=12$$

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.

$$x+2y=12,9x-2y=12$$

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

$$x+2y=12$$

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

$$x=-2y+12$$

Whakakapia te -2y+12 mō te x ki tērā atu whārite, 9x-2y=12.

$$9\left(-2y+12\right)-2y=12$$

Whakareatia 9 ki te -2y+12.

$$-18y+108-2y=12$$

Tāpiri -18y ki te -2y.

$$-20y+108=12$$

Me tango 108 mai i ngā taha e rua o te whārite.

$$-20y=-96$$

Whakawehea ngā taha e rua ki te -20.

$$y=\frac{24}{5}$$

Whakaurua te \frac{24}{5} mō y ki x=-2y+12. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-2\times \frac{24}{5}+12$$

Whakareatia -2 ki te \frac{24}{5}.

$$x=-\frac{48}{5}+12$$

Tāpiri 12 ki te -\frac{48}{5}.

$$x=\frac{12}{5}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{12}{5},y=\frac{24}{5}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$9x-2y=12$$

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

$$x+2y=12,9x-2y=12$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&2\\9&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}12\\12\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&2\\9&-2\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&2\\9&-2\end{matrix}\right))\left(\begin{matrix}1&2\\9&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\9&-2\end{matrix}\right))\left(\begin{matrix}12\\12\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\9&-2\end{matrix}\right))\left(\begin{matrix}12\\12\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\9&-2\end{matrix}\right))\left(\begin{matrix}12\\12\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{-2-2\times 9}&-\frac{2}{-2-2\times 9}\\-\frac{9}{-2-2\times 9}&\frac{1}{-2-2\times 9}\end{matrix}\right)\left(\begin{matrix}12\\12\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{10}&\frac{1}{10}\\\frac{9}{20}&-\frac{1}{20}\end{matrix}\right)\left(\begin{matrix}12\\12\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{10}\times 12+\frac{1}{10}\times 12\\\frac{9}{20}\times 12-\frac{1}{20}\times 12\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{12}{5}\\\frac{24}{5}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{12}{5},y=\frac{24}{5}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$9x-2y=12$$

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.

$$x+2y=12,9x-2y=12$$

Kia ōrite ai a x me 9x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 9 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$9x+9\times 2y=9\times 12,9x-2y=12$$

Whakarūnātia.

$$9x+18y=108,9x-2y=12$$

Me tango 9x-2y=12 mai i 9x+18y=108 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$9x-9x+18y+2y=108-12$$

Tāpiri 9x ki te -9x. Ka whakakore atu ngā kupu 9x me -9x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$18y+2y=108-12$$

Tāpiri 18y ki te 2y.

$$20y=108-12$$

Tāpiri 108 ki te -12.

$$20y=96$$

Whakawehea ngā taha e rua ki te 20.

$$y=\frac{24}{5}$$

Whakaurua te \frac{24}{5} mō y ki 9x-2y=12. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$9x-2\times \frac{24}{5}=12$$

Whakareatia -2 ki te \frac{24}{5}.

$$9x-\frac{48}{5}=12$$

Me tāpiri \frac{48}{5} ki ngā taha e rua o te whārite.

$$9x=\frac{108}{5}$$

Whakawehea ngā taha e rua ki te 9.

$$x=\frac{12}{5}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{12}{5},y=\frac{24}{5}$$