Whakaoti i te {l}{y-2x=4}{2x+3y=28}

$$\left\{ \begin{array} { l } { y - 2 x = 4 } \\ { 2 x + 3 y = 28 } \end{array} \right\}$$

Whakaoti mō y, x

$$x=2$$

$$y=8$$

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.

$$y-2x=4,3y+2x=28$$

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

$$y-2x=4$$

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

$$y=2x+4$$

Whakakapia te 4+2x mō te y ki tērā atu whārite, 3y+2x=28.

$$3\left(2x+4\right)+2x=28$$

Whakareatia 3 ki te 4+2x.

$$6x+12+2x=28$$

Tāpiri 6x ki te 2x.

$$8x+12=28$$

Me tango 12 mai i ngā taha e rua o te whārite.

$$8x=16$$

Whakawehea ngā taha e rua ki te 8.

$$x=2$$

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

$$y=2\times 2+4$$

Whakareatia 2 ki te 2.

$$y=4+4$$

Tāpiri 4 ki te 4.

$$y=8$$

Kua oti te pūnaha te whakatau.

$$y=8,x=2$$

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.

$$y-2x=4,3y+2x=28$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-2\\3&2\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}4\\28\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&-2\\3&2\end{matrix}\right))\left(\begin{matrix}1&-2\\3&2\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\3&2\end{matrix}\right))\left(\begin{matrix}4\\28\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\3&2\end{matrix}\right))\left(\begin{matrix}4\\28\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\3&2\end{matrix}\right))\left(\begin{matrix}4\\28\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{2}{2-\left(-2\times 3\right)}&-\frac{-2}{2-\left(-2\times 3\right)}\\-\frac{3}{2-\left(-2\times 3\right)}&\frac{1}{2-\left(-2\times 3\right)}\end{matrix}\right)\left(\begin{matrix}4\\28\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}&\frac{1}{4}\\-\frac{3}{8}&\frac{1}{8}\end{matrix}\right)\left(\begin{matrix}4\\28\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}\times 4+\frac{1}{4}\times 28\\-\frac{3}{8}\times 4+\frac{1}{8}\times 28\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}8\\2\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=8,x=2$$

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.

$$y-2x=4,3y+2x=28$$

Kia ōrite ai a y me 3y, 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 1.

$$3y+3\left(-2\right)x=3\times 4,3y+2x=28$$

Whakarūnātia.

$$3y-6x=12,3y+2x=28$$

Me tango 3y+2x=28 mai i 3y-6x=12 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3y-3y-6x-2x=12-28$$

Tāpiri 3y ki te -3y. Ka whakakore atu ngā kupu 3y me -3y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-6x-2x=12-28$$

Tāpiri -6x ki te -2x.

$$-8x=12-28$$

Tāpiri 12 ki te -28.

$$-8x=-16$$

Whakawehea ngā taha e rua ki te -8.

$$x=2$$

Whakaurua te 2 mō x ki 3y+2x=28. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$3y+2\times 2=28$$

Whakareatia 2 ki te 2.

$$3y+4=28$$

Me tango 4 mai i ngā taha e rua o te whārite.

$$3y=24$$

Whakawehea ngā taha e rua ki te 3.

$$y=8$$

Kua oti te pūnaha te whakatau.

$$y=8,x=2$$