Whakaoti i te {l}{y=2x-4}{x+2y-1=0}

$$\left. \begin{array} { l } { y = 2 x - 4 } \\ { x + 2 y - 1 = 0 } \end{array} \right.$$

Whakaoti mō y, x

$$x = \frac{9}{5} = 1\frac{4}{5} = 1.8$$

$$y=-\frac{2}{5}=-0.4$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

$$y-2x=-4$$

Whakaarohia te whārite tuarua. Me tāpiri te 1 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$x+2y=1$$

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.

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

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, 2y+x=1.

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

Whakareatia 2 ki te -4+2x.

$$4x-8+x=1$$

Tāpiri 4x ki te x.

$$5x-8=1$$

Me tāpiri 8 ki ngā taha e rua o te whārite.

$$5x=9$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{9}{5}$$

Whakaurua te \frac{9}{5} 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.

$$y=2\times \frac{9}{5}-4$$

Whakareatia 2 ki te \frac{9}{5}.

$$y=\frac{18}{5}-4$$

Tāpiri -4 ki te \frac{18}{5}.

$$y=-\frac{2}{5}$$

Kua oti te pūnaha te whakatau.

$$y=-\frac{2}{5},x=\frac{9}{5}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$y-2x=-4$$

Whakaarohia te whārite tuarua. Me tāpiri te 1 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$x+2y=1$$

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

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{5}\\\frac{9}{5}\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=-\frac{2}{5},x=\frac{9}{5}$$

Ngā Upane e Whakamahi i te Tangohanga

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

$$y-2x=-4$$

Whakaarohia te whārite tuarua. Me tāpiri te 1 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$x+2y=1$$

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.

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

Kia ōrite ai a y me 2y, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

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

Whakarūnātia.

$$2y-4x=-8,2y+x=1$$

Me tango 2y+x=1 mai i 2y-4x=-8 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2y-2y-4x-x=-8-1$$

Tāpiri 2y ki te -2y. Ka whakakore atu ngā kupu 2y me -2y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-4x-x=-8-1$$

Tāpiri -4x ki te -x.

$$-5x=-8-1$$

Tāpiri -8 ki te -1.

$$-5x=-9$$

Whakawehea ngā taha e rua ki te -5.

$$x=\frac{9}{5}$$

Whakaurua te \frac{9}{5} mō x ki 2y+x=1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$2y+\frac{9}{5}=1$$

Me tango \frac{9}{5} mai i ngā taha e rua o te whārite.

$$2y=-\frac{4}{5}$$

Whakawehea ngā taha e rua ki te 2.

$$y=-\frac{2}{5}$$

Kua oti te pūnaha te whakatau.

$$y=-\frac{2}{5},x=\frac{9}{5}$$