Whakaoti i te {l}{3x-y=-1}{-x+2y=7}

$$\left\{ \begin{array} { l } { 3 x - y = - 1 } \\ { - x + 2 y = 7 } \end{array} \right.$$

Whakaoti mō x, y

$$x=1$$

$$y=4$$

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.

$$3x-y=-1,-x+2y=7$$

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

$$3x-y=-1$$

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

$$3x=y-1$$

Whakawehea ngā taha e rua ki te 3.

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

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

$$x=\frac{1}{3}y-\frac{1}{3}$$

Whakakapia te \frac{-1+y}{3} mō te x ki tērā atu whārite, -x+2y=7.

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

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

$$-\frac{1}{3}y+\frac{1}{3}+2y=7$$

Tāpiri -\frac{y}{3} ki te 2y.

$$\frac{5}{3}y+\frac{1}{3}=7$$

Me tango \frac{1}{3} mai i ngā taha e rua o te whārite.

$$\frac{5}{3}y=\frac{20}{3}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{5}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=4$$

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

$$x=\frac{1}{3}\times 4-\frac{1}{3}$$

Whakareatia \frac{1}{3} ki te 4.

$$x=\frac{4-1}{3}$$

Tāpiri -\frac{1}{3} ki te \frac{4}{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.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=4$$

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.

$$3x-y=-1,-x+2y=7$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=1,y=4$$

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.

$$3x-y=-1,-x+2y=7$$

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

$$-3x-\left(-y\right)=-\left(-1\right),3\left(-1\right)x+3\times 2y=3\times 7$$

Whakarūnātia.

$$-3x+y=1,-3x+6y=21$$

Me tango -3x+6y=21 mai i -3x+y=1 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-3x+3x+y-6y=1-21$$

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.

$$y-6y=1-21$$

Tāpiri y ki te -6y.

$$-5y=1-21$$

Tāpiri 1 ki te -21.

$$-5y=-20$$

Whakawehea ngā taha e rua ki te -5.

$$y=4$$

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

$$-x+2\times 4=7$$

Whakareatia 2 ki te 4.

$$-x+8=7$$

Me tango 8 mai i ngā taha e rua o te whārite.

$$-x=-1$$

Whakawehea ngā taha e rua ki te -1.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=4$$