Whakaoti i te {l}{y-2x=4}{7x-y=1}

$$\left\{ \begin{array} { l } { y - 2 x = 4 } \\ { 7 x - y = 1 } \end{array} \right.$$

Whakaoti mō y, x

$$x=1$$

$$y=6$$

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.

$$y-2x=4,-y+7x=1$$

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, -y+7x=1.

$$-\left(2x+4\right)+7x=1$$

Whakareatia -1 ki te 4+2x.

$$-2x-4+7x=1$$

Tāpiri -2x ki te 7x.

$$5x-4=1$$

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

$$5x=5$$

Whakawehea ngā taha e rua ki te 5.

$$x=1$$

Whakaurua te 1 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.

$$y=2+4$$

Tāpiri 4 ki te 2.

$$y=6$$

Kua oti te pūnaha te whakatau.

$$y=6,x=1$$

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.

$$y-2x=4,-y+7x=1$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-2\\-1&7\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}4\\1\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-2\\-1&7\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-2\\-1&7\end{matrix}\right))\left(\begin{matrix}1&-2\\-1&7\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\-1&7\end{matrix}\right))\left(\begin{matrix}4\\1\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\-1&7\end{matrix}\right))\left(\begin{matrix}4\\1\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\-1&7\end{matrix}\right))\left(\begin{matrix}4\\1\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{7}{7-\left(-2\left(-1\right)\right)}&-\frac{-2}{7-\left(-2\left(-1\right)\right)}\\-\frac{-1}{7-\left(-2\left(-1\right)\right)}&\frac{1}{7-\left(-2\left(-1\right)\right)}\end{matrix}\right)\left(\begin{matrix}4\\1\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{7}{5}&\frac{2}{5}\\\frac{1}{5}&\frac{1}{5}\end{matrix}\right)\left(\begin{matrix}4\\1\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{7}{5}\times 4+\frac{2}{5}\\\frac{1}{5}\times 4+\frac{1}{5}\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}6\\1\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=6,x=1$$

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.

$$y-2x=4,-y+7x=1$$

Kia ōrite ai a y me -y, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -1 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$-y-\left(-2x\right)=-4,-y+7x=1$$

Whakarūnātia.

$$-y+2x=-4,-y+7x=1$$

Me tango -y+7x=1 mai i -y+2x=-4 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-y+y+2x-7x=-4-1$$

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

$$2x-7x=-4-1$$

Tāpiri 2x ki te -7x.

$$-5x=-4-1$$

Tāpiri -4 ki te -1.

$$-5x=-5$$

Whakawehea ngā taha e rua ki te -5.

$$x=1$$

Whakaurua te 1 mō x ki -y+7x=1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$-y+7=1$$

Me tango 7 mai i ngā taha e rua o te whārite.

$$-y=-6$$

Whakawehea ngā taha e rua ki te -1.

$$y=6$$

Kua oti te pūnaha te whakatau.

$$y=6,x=1$$