Whakaoti i te x+2y=8quad,quad3x+y=9

$$x + 2 y = 8 \quad , \quad 3 x + y = 9$$

Whakaoti mō x, y

$$x=2$$

$$y=3$$

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.

$$x+2y=8,3x+y=9$$

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

$$x+2y=8$$

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

$$x=-2y+8$$

Whakakapia te -2y+8 mō te x ki tērā atu whārite, 3x+y=9.

$$3\left(-2y+8\right)+y=9$$

Whakareatia 3 ki te -2y+8.

$$-6y+24+y=9$$

Tāpiri -6y ki te y.

$$-5y+24=9$$

Me tango 24 mai i ngā taha e rua o te whārite.

$$-5y=-15$$

Whakawehea ngā taha e rua ki te -5.

$$y=3$$

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

$$x=-2\times 3+8$$

Whakareatia -2 ki te 3.

$$x=-6+8$$

Tāpiri 8 ki te -6.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=3$$

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.

$$x+2y=8,3x+y=9$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&2\\3&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}8\\9\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&2\\3&1\end{matrix}\right))\left(\begin{matrix}1&2\\3&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\3&1\end{matrix}\right))\left(\begin{matrix}8\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\3&1\end{matrix}\right))\left(\begin{matrix}8\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\3&1\end{matrix}\right))\left(\begin{matrix}8\\9\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{1-2\times 3}&-\frac{2}{1-2\times 3}\\-\frac{3}{1-2\times 3}&\frac{1}{1-2\times 3}\end{matrix}\right)\left(\begin{matrix}8\\9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{5}&\frac{2}{5}\\\frac{3}{5}&-\frac{1}{5}\end{matrix}\right)\left(\begin{matrix}8\\9\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{5}\times 8+\frac{2}{5}\times 9\\\frac{3}{5}\times 8-\frac{1}{5}\times 9\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=2,y=3$$

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.

$$x+2y=8,3x+y=9$$

Kia ōrite ai a x me 3x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 3 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$3x+3\times 2y=3\times 8,3x+y=9$$

Whakarūnātia.

$$3x+6y=24,3x+y=9$$

Me tango 3x+y=9 mai i 3x+6y=24 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3x-3x+6y-y=24-9$$

Tāpiri 3x ki te -3x. Ka whakakore atu ngā kupu 3x me -3x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$6y-y=24-9$$

Tāpiri 6y ki te -y.

$$5y=24-9$$

Tāpiri 24 ki te -9.

$$5y=15$$

Whakawehea ngā taha e rua ki te 5.

$$y=3$$

Whakaurua te 3 mō y ki 3x+y=9. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x+3=9$$

Me tango 3 mai i ngā taha e rua o te whārite.

$$3x=6$$

Whakawehea ngā taha e rua ki te 3.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=3$$