Whakaoti i te {l}{5x+1y=2}{2x-5y=2}

$$\left\{ \begin{array} { l } { 5 x + 1 y = 2 } \\ { 2 x - 5 y = 2 } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{4}{9}\approx 0.444444444$$

$$y=-\frac{2}{9}\approx -0.222222222$$

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.

$$5x+y=2,2x-5y=2$$

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

$$5x+y=2$$

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

$$5x=-y+2$$

Whakawehea ngā taha e rua ki te 5.

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

Whakareatia \frac{1}{5} ki te -y+2.

$$x=-\frac{1}{5}y+\frac{2}{5}$$

Whakakapia te \frac{-y+2}{5} mō te x ki tērā atu whārite, 2x-5y=2.

$$2\left(-\frac{1}{5}y+\frac{2}{5}\right)-5y=2$$

Whakareatia 2 ki te \frac{-y+2}{5}.

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

Tāpiri -\frac{2y}{5} ki te -5y.

$$-\frac{27}{5}y+\frac{4}{5}=2$$

Me tango \frac{4}{5} mai i ngā taha e rua o te whārite.

$$-\frac{27}{5}y=\frac{6}{5}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{27}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-\frac{2}{9}$$

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

$$x=-\frac{1}{5}\left(-\frac{2}{9}\right)+\frac{2}{5}$$

Whakareatia -\frac{1}{5} ki te -\frac{2}{9} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=\frac{2}{45}+\frac{2}{5}$$

Tāpiri \frac{2}{5} ki te \frac{2}{45} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=\frac{4}{9}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{4}{9},y=-\frac{2}{9}$$

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.

$$5x+y=2,2x-5y=2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&1\\2&-5\end{matrix}\right).

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

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&1\\2&-5\end{matrix}\right))\left(\begin{matrix}2\\2\end{matrix}\right)$$

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{27}&\frac{1}{27}\\\frac{2}{27}&-\frac{5}{27}\end{matrix}\right)\left(\begin{matrix}2\\2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{27}\times 2+\frac{1}{27}\times 2\\\frac{2}{27}\times 2-\frac{5}{27}\times 2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{9}\\-\frac{2}{9}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{4}{9},y=-\frac{2}{9}$$

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.

$$5x+y=2,2x-5y=2$$

Kia ōrite ai a 5x me 2x, 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 5.

$$2\times 5x+2y=2\times 2,5\times 2x+5\left(-5\right)y=5\times 2$$

Whakarūnātia.

$$10x+2y=4,10x-25y=10$$

Me tango 10x-25y=10 mai i 10x+2y=4 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$10x-10x+2y+25y=4-10$$

Tāpiri 10x ki te -10x. Ka whakakore atu ngā kupu 10x me -10x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$2y+25y=4-10$$

Tāpiri 2y ki te 25y.

$$27y=4-10$$

Tāpiri 4 ki te -10.

$$27y=-6$$

Whakawehea ngā taha e rua ki te 27.

$$y=-\frac{2}{9}$$

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

$$2x-5\left(-\frac{2}{9}\right)=2$$

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

$$2x+\frac{10}{9}=2$$

Me tango \frac{10}{9} mai i ngā taha e rua o te whārite.

$$2x=\frac{8}{9}$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{4}{9}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{4}{9},y=-\frac{2}{9}$$