Whakaoti i te 3x-12y=15;quad2x-9y=11

$$3 x - 12 y = 15 ; \quad 2 x - 9 y = 11$$

Whakaoti mō x, y

$$x=1$$

$$y=-1$$

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.

$$3x-12y=15,2x-9y=11$$

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

$$3x-12y=15$$

Me tāpiri 12y ki ngā taha e rua o te whārite.

$$3x=12y+15$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(12y+15\right)$$

Whakareatia \frac{1}{3} ki te 12y+15.

$$x=4y+5$$

Whakakapia te 4y+5 mō te x ki tērā atu whārite, 2x-9y=11.

$$2\left(4y+5\right)-9y=11$$

Whakareatia 2 ki te 4y+5.

$$8y+10-9y=11$$

Tāpiri 8y ki te -9y.

$$-y+10=11$$

Me tango 10 mai i ngā taha e rua o te whārite.

$$-y=1$$

Whakawehea ngā taha e rua ki te -1.

$$y=-1$$

Whakaurua te -1 mō y ki x=4y+5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=4\left(-1\right)+5$$

Whakareatia 4 ki te -1.

$$x=-4+5$$

Tāpiri 5 ki te -4.

$$x=1$$

Kua oti te pūnaha te whakatau.

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

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.

$$3x-12y=15,2x-9y=11$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&-12\\2&-9\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}15\\11\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&-12\\2&-9\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&-12\\2&-9\end{matrix}\right))\left(\begin{matrix}3&-12\\2&-9\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-12\\2&-9\end{matrix}\right))\left(\begin{matrix}15\\11\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-12\\2&-9\end{matrix}\right))\left(\begin{matrix}15\\11\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-12\\2&-9\end{matrix}\right))\left(\begin{matrix}15\\11\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{9}{3\left(-9\right)-\left(-12\times 2\right)}&-\frac{-12}{3\left(-9\right)-\left(-12\times 2\right)}\\-\frac{2}{3\left(-9\right)-\left(-12\times 2\right)}&\frac{3}{3\left(-9\right)-\left(-12\times 2\right)}\end{matrix}\right)\left(\begin{matrix}15\\11\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3&-4\\\frac{2}{3}&-1\end{matrix}\right)\left(\begin{matrix}15\\11\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3\times 15-4\times 11\\\frac{2}{3}\times 15-11\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

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

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.

$$3x-12y=15,2x-9y=11$$

Kia ōrite ai a 3x me 2x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$2\times 3x+2\left(-12\right)y=2\times 15,3\times 2x+3\left(-9\right)y=3\times 11$$

Whakarūnātia.

$$6x-24y=30,6x-27y=33$$

Me tango 6x-27y=33 mai i 6x-24y=30 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x-24y+27y=30-33$$

Tāpiri 6x ki te -6x. Ka whakakore atu ngā kupu 6x me -6x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-24y+27y=30-33$$

Tāpiri -24y ki te 27y.

$$3y=30-33$$

Tāpiri 30 ki te -33.

$$3y=-3$$

Whakawehea ngā taha e rua ki te 3.

$$y=-1$$

Whakaurua te -1 mō y ki 2x-9y=11. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x-9\left(-1\right)=11$$

Whakareatia -9 ki te -1.

$$2x+9=11$$

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

$$2x=2$$

Whakawehea ngā taha e rua ki te 2.

$$x=1$$

Kua oti te pūnaha te whakatau.

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