Whakaoti i te {c}{x-2y=9}{3x+4y=7}

$$\left\{ \begin{array} { c } { x - 2 y = 9 } \\ { 3 x + 4 y = 7 } \end{array} \right.$$

Whakaoti mō x, y

$$x=5$$

$$y=-2$$

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.

$$x-2y=9,3x+4y=7$$

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

$$x-2y=9$$

Me tāpiri 2y ki ngā taha e rua o te whārite.

$$x=2y+9$$

Whakakapia te 2y+9 mō te x ki tērā atu whārite, 3x+4y=7.

$$3\left(2y+9\right)+4y=7$$

Whakareatia 3 ki te 2y+9.

$$6y+27+4y=7$$

Tāpiri 6y ki te 4y.

$$10y+27=7$$

Me tango 27 mai i ngā taha e rua o te whārite.

$$10y=-20$$

Whakawehea ngā taha e rua ki te 10.

$$y=-2$$

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

$$x=2\left(-2\right)+9$$

Whakareatia 2 ki te -2.

$$x=-4+9$$

Tāpiri 9 ki te -4.

$$x=5$$

Kua oti te pūnaha te whakatau.

$$x=5,y=-2$$

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.

$$x-2y=9,3x+4y=7$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-2\\3&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}9\\7\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-2\\3&4\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-2\\3&4\end{matrix}\right))\left(\begin{matrix}1&-2\\3&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\3&4\end{matrix}\right))\left(\begin{matrix}9\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\3&4\end{matrix}\right))\left(\begin{matrix}9\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\3&4\end{matrix}\right))\left(\begin{matrix}9\\7\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{4-\left(-2\times 3\right)}&-\frac{-2}{4-\left(-2\times 3\right)}\\-\frac{3}{4-\left(-2\times 3\right)}&\frac{1}{4-\left(-2\times 3\right)}\end{matrix}\right)\left(\begin{matrix}9\\7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{5}&\frac{1}{5}\\-\frac{3}{10}&\frac{1}{10}\end{matrix}\right)\left(\begin{matrix}9\\7\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{5}\times 9+\frac{1}{5}\times 7\\-\frac{3}{10}\times 9+\frac{1}{10}\times 7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\\-2\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=5,y=-2$$

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.

$$x-2y=9,3x+4y=7$$

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.

$$3x+3\left(-2\right)y=3\times 9,3x+4y=7$$

Whakarūnātia.

$$3x-6y=27,3x+4y=7$$

Me tango 3x+4y=7 mai i 3x-6y=27 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3x-3x-6y-4y=27-7$$

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.

$$-6y-4y=27-7$$

Tāpiri -6y ki te -4y.

$$-10y=27-7$$

Tāpiri 27 ki te -7.

$$-10y=20$$

Whakawehea ngā taha e rua ki te -10.

$$y=-2$$

Whakaurua te -2 mō y ki 3x+4y=7. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x+4\left(-2\right)=7$$

Whakareatia 4 ki te -2.

$$3x-8=7$$

Me tāpiri 8 ki ngā taha e rua o te whārite.

$$3x=15$$

Whakawehea ngā taha e rua ki te 3.

$$x=5$$

Kua oti te pūnaha te whakatau.

$$x=5,y=-2$$