Whakaoti i te {l}{12x-4y=-4}{3x+8y=17}

$$\left. \begin{array}{l}{ 12 x - 4 y = - 4 }\\{ 3 x + 8 y = 17 }\end{array} \right.$$

Whakaoti mō x, y

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

$$y=2$$

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.

$$12x-4y=-4,3x+8y=17$$

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

$$12x-4y=-4$$

Me tāpiri 4y ki ngā taha e rua o te whārite.

$$12x=4y-4$$

Whakawehea ngā taha e rua ki te 12.

$$x=\frac{1}{12}\left(4y-4\right)$$

Whakareatia \frac{1}{12} ki te -4+4y.

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

Whakakapia te \frac{-1+y}{3} mō te x ki tērā atu whārite, 3x+8y=17.

$$3\left(\frac{1}{3}y-\frac{1}{3}\right)+8y=17$$

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

$$y-1+8y=17$$

Tāpiri y ki te 8y.

$$9y-1=17$$

Me tāpiri 1 ki ngā taha e rua o te whārite.

$$9y=18$$

Whakawehea ngā taha e rua ki te 9.

$$y=2$$

Whakaurua te 2 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.

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

Whakareatia \frac{1}{3} ki te 2.

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

Tāpiri -\frac{1}{3} ki te \frac{2}{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.

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

Kua oti te pūnaha te whakatau.

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

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.

$$12x-4y=-4,3x+8y=17$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}12&-4\\3&8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-4\\17\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}12&-4\\3&8\end{matrix}\right).

$$inverse(\left(\begin{matrix}12&-4\\3&8\end{matrix}\right))\left(\begin{matrix}12&-4\\3&8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}12&-4\\3&8\end{matrix}\right))\left(\begin{matrix}-4\\17\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}12&-4\\3&8\end{matrix}\right))\left(\begin{matrix}-4\\17\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}12&-4\\3&8\end{matrix}\right))\left(\begin{matrix}-4\\17\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{8}{12\times 8-\left(-4\times 3\right)}&-\frac{-4}{12\times 8-\left(-4\times 3\right)}\\-\frac{3}{12\times 8-\left(-4\times 3\right)}&\frac{12}{12\times 8-\left(-4\times 3\right)}\end{matrix}\right)\left(\begin{matrix}-4\\17\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{27}&\frac{1}{27}\\-\frac{1}{36}&\frac{1}{9}\end{matrix}\right)\left(\begin{matrix}-4\\17\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{27}\left(-4\right)+\frac{1}{27}\times 17\\-\frac{1}{36}\left(-4\right)+\frac{1}{9}\times 17\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

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

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.

$$12x-4y=-4,3x+8y=17$$

Kia ōrite ai a 12x 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 12.

$$3\times 12x+3\left(-4\right)y=3\left(-4\right),12\times 3x+12\times 8y=12\times 17$$

Whakarūnātia.

$$36x-12y=-12,36x+96y=204$$

Me tango 36x+96y=204 mai i 36x-12y=-12 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$36x-36x-12y-96y=-12-204$$

Tāpiri 36x ki te -36x. Ka whakakore atu ngā kupu 36x me -36x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-12y-96y=-12-204$$

Tāpiri -12y ki te -96y.

$$-108y=-12-204$$

Tāpiri -12 ki te -204.

$$-108y=-216$$

Whakawehea ngā taha e rua ki te -108.

$$y=2$$

Whakaurua te 2 mō y ki 3x+8y=17. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x+8\times 2=17$$

Whakareatia 8 ki te 2.

$$3x+16=17$$

Me tango 16 mai i ngā taha e rua o te whārite.

$$3x=1$$

Whakawehea ngā taha e rua ki te 3.

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

Kua oti te pūnaha te whakatau.

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