Whakaoti i te -2x-y=10text{and}3x-4y=-4

$$- 2 x - y = 10 \text { and } 3 x - 4 y = - 4$$

Whakaoti mō x, y

$$x=-4$$

$$y=-2$$

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.

$$-2x-y=10,3x-4y=-4$$

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

$$-2x-y=10$$

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

$$-2x=y+10$$

Whakawehea ngā taha e rua ki te -2.

$$x=-\frac{1}{2}\left(y+10\right)$$

Whakareatia -\frac{1}{2} ki te y+10.

$$x=-\frac{1}{2}y-5$$

Whakakapia te -\frac{y}{2}-5 mō te x ki tērā atu whārite, 3x-4y=-4.

$$3\left(-\frac{1}{2}y-5\right)-4y=-4$$

Whakareatia 3 ki te -\frac{y}{2}-5.

$$-\frac{3}{2}y-15-4y=-4$$

Tāpiri -\frac{3y}{2} ki te -4y.

$$-\frac{11}{2}y-15=-4$$

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

$$-\frac{11}{2}y=11$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{11}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-2$$

Whakaurua te -2 mō y ki x=-\frac{1}{2}y-5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{2}\left(-2\right)-5$$

Whakareatia -\frac{1}{2} ki te -2.

$$x=1-5$$

Tāpiri -5 ki te 1.

$$x=-4$$

Kua oti te pūnaha te whakatau.

$$x=-4,y=-2$$

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.

$$-2x-y=10,3x-4y=-4$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-2&-1\\3&-4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}10\\-4\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-2&-1\\3&-4\end{matrix}\right).

$$inverse(\left(\begin{matrix}-2&-1\\3&-4\end{matrix}\right))\left(\begin{matrix}-2&-1\\3&-4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-2&-1\\3&-4\end{matrix}\right))\left(\begin{matrix}10\\-4\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-2&-1\\3&-4\end{matrix}\right))\left(\begin{matrix}10\\-4\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-2&-1\\3&-4\end{matrix}\right))\left(\begin{matrix}10\\-4\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{-2\left(-4\right)-\left(-3\right)}&-\frac{-1}{-2\left(-4\right)-\left(-3\right)}\\-\frac{3}{-2\left(-4\right)-\left(-3\right)}&-\frac{2}{-2\left(-4\right)-\left(-3\right)}\end{matrix}\right)\left(\begin{matrix}10\\-4\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{11}&\frac{1}{11}\\-\frac{3}{11}&-\frac{2}{11}\end{matrix}\right)\left(\begin{matrix}10\\-4\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{11}\times 10+\frac{1}{11}\left(-4\right)\\-\frac{3}{11}\times 10-\frac{2}{11}\left(-4\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=-4,y=-2$$

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.

$$-2x-y=10,3x-4y=-4$$

Kia ōrite ai a -2x 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 -2.

$$3\left(-2\right)x+3\left(-1\right)y=3\times 10,-2\times 3x-2\left(-4\right)y=-2\left(-4\right)$$

Whakarūnātia.

$$-6x-3y=30,-6x+8y=8$$

Me tango -6x+8y=8 mai i -6x-3y=30 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-6x+6x-3y-8y=30-8$$

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.

$$-3y-8y=30-8$$

Tāpiri -3y ki te -8y.

$$-11y=30-8$$

Tāpiri 30 ki te -8.

$$-11y=22$$

Whakawehea ngā taha e rua ki te -11.

$$y=-2$$

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

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

Whakareatia -4 ki te -2.

$$3x+8=-4$$

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

$$3x=-12$$

Whakawehea ngā taha e rua ki te 3.

$$x=-4$$

Kua oti te pūnaha te whakatau.

$$x=-4,y=-2$$