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

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

Whakaoti mō x, y

$$x=2$$

$$y=-1$$

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.

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

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

$$-3x+y=-7$$

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

$$-3x=-y-7$$

Whakawehea ngā taha e rua ki te -3.

$$x=-\frac{1}{3}\left(-y-7\right)$$

Whakareatia -\frac{1}{3} ki te -y-7.

$$x=\frac{1}{3}y+\frac{7}{3}$$

Whakakapia te \frac{7+y}{3} mō te x ki tērā atu whārite, 3x+2y=4.

$$3\left(\frac{1}{3}y+\frac{7}{3}\right)+2y=4$$

Whakareatia 3 ki te \frac{7+y}{3}.

$$y+7+2y=4$$

Tāpiri y ki te 2y.

$$3y+7=4$$

Me tango 7 mai i ngā taha e rua o te whārite.

$$3y=-3$$

Whakawehea ngā taha e rua ki te 3.

$$y=-1$$

Whakaurua te -1 mō y ki x=\frac{1}{3}y+\frac{7}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{1}{3}\left(-1\right)+\frac{7}{3}$$

Whakareatia \frac{1}{3} ki te -1.

$$x=\frac{-1+7}{3}$$

Tāpiri \frac{7}{3} ki te -\frac{1}{3} 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.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=-1$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=2,y=-1$$

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.

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

Kia ōrite ai a -3x 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 -3.

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

Whakarūnātia.

$$-9x+3y=-21,-9x-6y=-12$$

Me tango -9x-6y=-12 mai i -9x+3y=-21 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-9x+9x+3y+6y=-21+12$$

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.

$$3y+6y=-21+12$$

Tāpiri 3y ki te 6y.

$$9y=-21+12$$

Tāpiri -21 ki te 12.

$$9y=-9$$

Whakawehea ngā taha e rua ki te 9.

$$y=-1$$

Whakaurua te -1 mō y ki 3x+2y=4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x+2\left(-1\right)=4$$

Whakareatia 2 ki te -1.

$$3x-2=4$$

Me tāpiri 2 ki 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.

$$x=2,y=-1$$