Whakaoti i te {l}{-2x+3y=9}{7x-9y=-31}

$$\left\{ \begin{array} { l } { - 2 x + 3 y = 9 } \\ { 7 x - 9 y = - 31 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-4$$

$$y=\frac{1}{3}\approx 0.333333333$$

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.

$$-2x+3y=9,7x-9y=-31$$

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

$$-2x+3y=9$$

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

$$-2x=-3y+9$$

Whakawehea ngā taha e rua ki te -2.

$$x=-\frac{1}{2}\left(-3y+9\right)$$

Whakareatia -\frac{1}{2} ki te -3y+9.

$$x=\frac{3}{2}y-\frac{9}{2}$$

Whakakapia te \frac{-9+3y}{2} mō te x ki tērā atu whārite, 7x-9y=-31.

$$7\left(\frac{3}{2}y-\frac{9}{2}\right)-9y=-31$$

Whakareatia 7 ki te \frac{-9+3y}{2}.

$$\frac{21}{2}y-\frac{63}{2}-9y=-31$$

Tāpiri \frac{21y}{2} ki te -9y.

$$\frac{3}{2}y-\frac{63}{2}=-31$$

Me tāpiri \frac{63}{2} ki ngā taha e rua o te whārite.

$$\frac{3}{2}y=\frac{1}{2}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{3}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

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

Whakaurua te \frac{1}{3} mō y ki x=\frac{3}{2}y-\frac{9}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{3}{2}\times \frac{1}{3}-\frac{9}{2}$$

Whakareatia \frac{3}{2} ki te \frac{1}{3} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=\frac{1-9}{2}$$

Tāpiri -\frac{9}{2} ki te \frac{1}{2} 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.

$$x=-4$$

Kua oti te pūnaha te whakatau.

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

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.

$$-2x+3y=9,7x-9y=-31$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-2&3\\7&-9\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}9\\-31\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-2&3\\7&-9\end{matrix}\right).

$$inverse(\left(\begin{matrix}-2&3\\7&-9\end{matrix}\right))\left(\begin{matrix}-2&3\\7&-9\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-2&3\\7&-9\end{matrix}\right))\left(\begin{matrix}9\\-31\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-2&3\\7&-9\end{matrix}\right))\left(\begin{matrix}9\\-31\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-2&3\\7&-9\end{matrix}\right))\left(\begin{matrix}9\\-31\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{9}{-2\left(-9\right)-3\times 7}&-\frac{3}{-2\left(-9\right)-3\times 7}\\-\frac{7}{-2\left(-9\right)-3\times 7}&-\frac{2}{-2\left(-9\right)-3\times 7}\end{matrix}\right)\left(\begin{matrix}9\\-31\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3&1\\\frac{7}{3}&\frac{2}{3}\end{matrix}\right)\left(\begin{matrix}9\\-31\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3\times 9-31\\\frac{7}{3}\times 9+\frac{2}{3}\left(-31\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

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

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.

$$-2x+3y=9,7x-9y=-31$$

Kia ōrite ai a -2x me 7x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 7 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te -2.

$$7\left(-2\right)x+7\times 3y=7\times 9,-2\times 7x-2\left(-9\right)y=-2\left(-31\right)$$

Whakarūnātia.

$$-14x+21y=63,-14x+18y=62$$

Me tango -14x+18y=62 mai i -14x+21y=63 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-14x+14x+21y-18y=63-62$$

Tāpiri -14x ki te 14x. Ka whakakore atu ngā kupu -14x me 14x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$21y-18y=63-62$$

Tāpiri 21y ki te -18y.

$$3y=63-62$$

Tāpiri 63 ki te -62.

$$3y=1$$

Whakawehea ngā taha e rua ki te 3.

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

Whakaurua te \frac{1}{3} mō y ki 7x-9y=-31. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$7x-9\times \frac{1}{3}=-31$$

Whakareatia -9 ki te \frac{1}{3}.

$$7x-3=-31$$

Me tāpiri 3 ki ngā taha e rua o te whārite.

$$7x=-28$$

Whakawehea ngā taha e rua ki te 7.

$$x=-4$$

Kua oti te pūnaha te whakatau.

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