Whakaoti i te {l}{y=2x}{x+3y=14}

$$\left\{ \begin{array} { l } { y = 2 x } \\ { x + 3 y = 14 } \end{array} \right.$$

Whakaoti mō y, x

$$x=2$$

$$y=4$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tangohia te 2x mai i ngā taha e rua.

$$y-2x=0$$

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.

$$y-2x=0,3y+x=14$$

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

$$y-2x=0$$

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

$$y=2x$$

Whakakapia te 2x mō te y ki tērā atu whārite, 3y+x=14.

$$3\times 2x+x=14$$

Whakareatia 3 ki te 2x.

$$6x+x=14$$

Tāpiri 6x ki te x.

$$7x=14$$

Whakawehea ngā taha e rua ki te 7.

$$x=2$$

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

$$y=2\times 2$$

Whakareatia 2 ki te 2.

$$y=4$$

Kua oti te pūnaha te whakatau.

$$y=4,x=2$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tangohia te 2x mai i ngā taha e rua.

$$y-2x=0$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$y-2x=0,3y+x=14$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-2\\3&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}0\\14\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&-2\\3&1\end{matrix}\right))\left(\begin{matrix}1&-2\\3&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\3&1\end{matrix}\right))\left(\begin{matrix}0\\14\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\3&1\end{matrix}\right))\left(\begin{matrix}0\\14\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\3&1\end{matrix}\right))\left(\begin{matrix}0\\14\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{1-\left(-2\times 3\right)}&-\frac{-2}{1-\left(-2\times 3\right)}\\-\frac{3}{1-\left(-2\times 3\right)}&\frac{1}{1-\left(-2\times 3\right)}\end{matrix}\right)\left(\begin{matrix}0\\14\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{7}&\frac{2}{7}\\-\frac{3}{7}&\frac{1}{7}\end{matrix}\right)\left(\begin{matrix}0\\14\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{2}{7}\times 14\\\frac{1}{7}\times 14\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa y me x.

$$y=4,x=2$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Tangohia te 2x mai i ngā taha e rua.

$$y-2x=0$$

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.

$$y-2x=0,3y+x=14$$

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.

$$3y+3\left(-2\right)x=0,3y+x=14$$

Whakarūnātia.

$$3y-6x=0,3y+x=14$$

Me tango 3y+x=14 mai i 3y-6x=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3y-3y-6x-x=-14$$

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.

$$-6x-x=-14$$

Tāpiri -6x ki te -x.

$$-7x=-14$$

Whakawehea ngā taha e rua ki te -7.

$$x=2$$

Whakaurua te 2 mō x ki 3y+x=14. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$3y+2=14$$

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

$$3y=12$$

Whakawehea ngā taha e rua ki te 3.

$$y=4$$

Kua oti te pūnaha te whakatau.

$$y=4,x=2$$