Whakaoti i te x-y=5text{and}5y-4x=7

$$x - y = 5 \text { and } 5 y - 4 x = 7$$

Whakaoti mō x, y

$$x=32$$

$$y=27$$

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.

$$x-y=5,-4x+5y=7$$

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

$$x-y=5$$

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

$$x=y+5$$

Whakakapia te y+5 mō te x ki tērā atu whārite, -4x+5y=7.

$$-4\left(y+5\right)+5y=7$$

Whakareatia -4 ki te y+5.

$$-4y-20+5y=7$$

Tāpiri -4y ki te 5y.

$$y-20=7$$

Me tāpiri 20 ki ngā taha e rua o te whārite.

$$y=27$$

Whakaurua te 27 mō y ki x=y+5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=27+5$$

Tāpiri 5 ki te 27.

$$x=32$$

Kua oti te pūnaha te whakatau.

$$x=32,y=27$$

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.

$$x-y=5,-4x+5y=7$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-1\\-4&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\\7\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-1\\-4&5\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-1\\-4&5\end{matrix}\right))\left(\begin{matrix}1&-1\\-4&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\-4&5\end{matrix}\right))\left(\begin{matrix}5\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\-4&5\end{matrix}\right))\left(\begin{matrix}5\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\-4&5\end{matrix}\right))\left(\begin{matrix}5\\7\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{5-\left(-\left(-4\right)\right)}&-\frac{-1}{5-\left(-\left(-4\right)\right)}\\-\frac{-4}{5-\left(-\left(-4\right)\right)}&\frac{1}{5-\left(-\left(-4\right)\right)}\end{matrix}\right)\left(\begin{matrix}5\\7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5&1\\4&1\end{matrix}\right)\left(\begin{matrix}5\\7\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\times 5+7\\4\times 5+7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}32\\27\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=32,y=27$$

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.

$$x-y=5,-4x+5y=7$$

Kia ōrite ai a x me -4x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -4 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$-4x-4\left(-1\right)y=-4\times 5,-4x+5y=7$$

Whakarūnātia.

$$-4x+4y=-20,-4x+5y=7$$

Me tango -4x+5y=7 mai i -4x+4y=-20 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-4x+4x+4y-5y=-20-7$$

Tāpiri -4x ki te 4x. Ka whakakore atu ngā kupu -4x me 4x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$4y-5y=-20-7$$

Tāpiri 4y ki te -5y.

$$-y=-20-7$$

Tāpiri -20 ki te -7.

$$-y=-27$$

Whakawehea ngā taha e rua ki te -1.

$$y=27$$

Whakaurua te 27 mō y ki -4x+5y=7. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-4x+5\times 27=7$$

Whakareatia 5 ki te 27.

$$-4x+135=7$$

Me tango 135 mai i ngā taha e rua o te whārite.

$$-4x=-128$$

Whakawehea ngā taha e rua ki te -4.

$$x=32$$

Kua oti te pūnaha te whakatau.

$$x=32,y=27$$