Whakaoti i te {c}{4x+2y=8}{16x-y=14}

$$\left. \begin{array} { c } { 4 x + 2 y = 8 } \\ { 16 x - y = 14 } \end{array} \right.$$

Whakaoti mō x, y

$$x=1$$

$$y=2$$

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.

$$4x+2y=8,16x-y=14$$

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

$$4x+2y=8$$

Me tango 2y mai i ngā taha e rua o te whārite.

$$4x=-2y+8$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{1}{4}\left(-2y+8\right)$$

Whakareatia \frac{1}{4} ki te -2y+8.

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

Whakakapia te -\frac{y}{2}+2 mō te x ki tērā atu whārite, 16x-y=14.

$$16\left(-\frac{1}{2}y+2\right)-y=14$$

Whakareatia 16 ki te -\frac{y}{2}+2.

$$-8y+32-y=14$$

Tāpiri -8y ki te -y.

$$-9y+32=14$$

Me tango 32 mai i 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}{2}y+2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

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

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

$$x=-1+2$$

Tāpiri 2 ki te -1.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=2$$

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.

$$4x+2y=8,16x-y=14$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&2\\16&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}8\\14\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4&2\\16&-1\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&2\\16&-1\end{matrix}\right))\left(\begin{matrix}4&2\\16&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&2\\16&-1\end{matrix}\right))\left(\begin{matrix}8\\14\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&2\\16&-1\end{matrix}\right))\left(\begin{matrix}8\\14\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&2\\16&-1\end{matrix}\right))\left(\begin{matrix}8\\14\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{4\left(-1\right)-2\times 16}&-\frac{2}{4\left(-1\right)-2\times 16}\\-\frac{16}{4\left(-1\right)-2\times 16}&\frac{4}{4\left(-1\right)-2\times 16}\end{matrix}\right)\left(\begin{matrix}8\\14\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{36}&\frac{1}{18}\\\frac{4}{9}&-\frac{1}{9}\end{matrix}\right)\left(\begin{matrix}8\\14\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{36}\times 8+\frac{1}{18}\times 14\\\frac{4}{9}\times 8-\frac{1}{9}\times 14\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=1,y=2$$

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.

$$4x+2y=8,16x-y=14$$

Kia ōrite ai a 4x me 16x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 16 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 4.

$$16\times 4x+16\times 2y=16\times 8,4\times 16x+4\left(-1\right)y=4\times 14$$

Whakarūnātia.

$$64x+32y=128,64x-4y=56$$

Me tango 64x-4y=56 mai i 64x+32y=128 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$64x-64x+32y+4y=128-56$$

Tāpiri 64x ki te -64x. Ka whakakore atu ngā kupu 64x me -64x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$32y+4y=128-56$$

Tāpiri 32y ki te 4y.

$$36y=128-56$$

Tāpiri 128 ki te -56.

$$36y=72$$

Whakawehea ngā taha e rua ki te 36.

$$y=2$$

Whakaurua te 2 mō y ki 16x-y=14. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$16x-2=14$$

Me tāpiri 2 ki ngā taha e rua o te whārite.

$$16x=16$$

Whakawehea ngā taha e rua ki te 16.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=2$$