Whakaoti i te {l}{-4x+9y=9}{x-3y=-6}

$$\left. \begin{array} { l } { - 4 x + 9 y = 9 } \\ { x - 3 y = - 6 } \end{array} \right.$$

Whakaoti mō x, y

$$x=9$$

$$y=5$$

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.

$$-4x+9y=9,x-3y=-6$$

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

$$-4x+9y=9$$

Me tango 9y mai i ngā taha e rua o te whārite.

$$-4x=-9y+9$$

Whakawehea ngā taha e rua ki te -4.

$$x=-\frac{1}{4}\left(-9y+9\right)$$

Whakareatia -\frac{1}{4} ki te -9y+9.

$$x=\frac{9}{4}y-\frac{9}{4}$$

Whakakapia te \frac{-9+9y}{4} mō te x ki tērā atu whārite, x-3y=-6.

$$\frac{9}{4}y-\frac{9}{4}-3y=-6$$

Tāpiri \frac{9y}{4} ki te -3y.

$$-\frac{3}{4}y-\frac{9}{4}=-6$$

Me tāpiri \frac{9}{4} ki ngā taha e rua o te whārite.

$$-\frac{3}{4}y=-\frac{15}{4}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{3}{4}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=5$$

Whakaurua te 5 mō y ki x=\frac{9}{4}y-\frac{9}{4}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{9}{4}\times 5-\frac{9}{4}$$

Whakareatia \frac{9}{4} ki te 5.

$$x=\frac{45-9}{4}$$

Tāpiri -\frac{9}{4} ki te \frac{45}{4} 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.

$$x=9$$

Kua oti te pūnaha te whakatau.

$$x=9,y=5$$

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.

$$-4x+9y=9,x-3y=-6$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-4&9\\1&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}9\\-6\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-4&9\\1&-3\end{matrix}\right).

$$inverse(\left(\begin{matrix}-4&9\\1&-3\end{matrix}\right))\left(\begin{matrix}-4&9\\1&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-4&9\\1&-3\end{matrix}\right))\left(\begin{matrix}9\\-6\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-4&9\\1&-3\end{matrix}\right))\left(\begin{matrix}9\\-6\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-4&9\\1&-3\end{matrix}\right))\left(\begin{matrix}9\\-6\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{-4\left(-3\right)-9}&-\frac{9}{-4\left(-3\right)-9}\\-\frac{1}{-4\left(-3\right)-9}&-\frac{4}{-4\left(-3\right)-9}\end{matrix}\right)\left(\begin{matrix}9\\-6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-1&-3\\-\frac{1}{3}&-\frac{4}{3}\end{matrix}\right)\left(\begin{matrix}9\\-6\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-9-3\left(-6\right)\\-\frac{1}{3}\times 9-\frac{4}{3}\left(-6\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}9\\5\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=9,y=5$$

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.

$$-4x+9y=9,x-3y=-6$$

Kia ōrite ai a -4x 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 -4.

$$-4x+9y=9,-4x-4\left(-3\right)y=-4\left(-6\right)$$

Whakarūnātia.

$$-4x+9y=9,-4x+12y=24$$

Me tango -4x+12y=24 mai i -4x+9y=9 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-4x+4x+9y-12y=9-24$$

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.

$$9y-12y=9-24$$

Tāpiri 9y ki te -12y.

$$-3y=9-24$$

Tāpiri 9 ki te -24.

$$-3y=-15$$

Whakawehea ngā taha e rua ki te -3.

$$y=5$$

Whakaurua te 5 mō y ki x-3y=-6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x-3\times 5=-6$$

Whakareatia -3 ki te 5.

$$x-15=-6$$

Me tāpiri 15 ki ngā taha e rua o te whārite.

$$x=9$$

Kua oti te pūnaha te whakatau.

$$x=9,y=5$$