Whakaoti i te {l}{5x-y=9}{2x+4y=8}

$$\left\{ \begin{array} { l } { 5 x - y = 9 } \\ { 2 x + 4 y = 8 } \end{array} \right.$$

Whakaoti mō x, y

$$x=2$$

$$y=1$$

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.

$$5x-y=9,2x+4y=8$$

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

$$5x-y=9$$

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

$$5x=y+9$$

Whakawehea ngā taha e rua ki te 5.

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

Whakareatia \frac{1}{5} ki te y+9.

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

Whakakapia te \frac{9+y}{5} mō te x ki tērā atu whārite, 2x+4y=8.

$$2\left(\frac{1}{5}y+\frac{9}{5}\right)+4y=8$$

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

$$\frac{2}{5}y+\frac{18}{5}+4y=8$$

Tāpiri \frac{2y}{5} ki te 4y.

$$\frac{22}{5}y+\frac{18}{5}=8$$

Me tango \frac{18}{5} mai i ngā taha e rua o te whārite.

$$\frac{22}{5}y=\frac{22}{5}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{22}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=1$$

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

$$x=\frac{1+9}{5}$$

Tāpiri \frac{9}{5} ki te \frac{1}{5} 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.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=1$$

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.

$$5x-y=9,2x+4y=8$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&-1\\2&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}9\\8\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&-1\\2&4\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&-1\\2&4\end{matrix}\right))\left(\begin{matrix}5&-1\\2&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-1\\2&4\end{matrix}\right))\left(\begin{matrix}9\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-1\\2&4\end{matrix}\right))\left(\begin{matrix}9\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-1\\2&4\end{matrix}\right))\left(\begin{matrix}9\\8\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{5\times 4-\left(-2\right)}&-\frac{-1}{5\times 4-\left(-2\right)}\\-\frac{2}{5\times 4-\left(-2\right)}&\frac{5}{5\times 4-\left(-2\right)}\end{matrix}\right)\left(\begin{matrix}9\\8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{11}&\frac{1}{22}\\-\frac{1}{11}&\frac{5}{22}\end{matrix}\right)\left(\begin{matrix}9\\8\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{11}\times 9+\frac{1}{22}\times 8\\-\frac{1}{11}\times 9+\frac{5}{22}\times 8\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=2,y=1$$

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.

$$5x-y=9,2x+4y=8$$

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.

$$2\times 5x+2\left(-1\right)y=2\times 9,5\times 2x+5\times 4y=5\times 8$$

Whakarūnātia.

$$10x-2y=18,10x+20y=40$$

Me tango 10x+20y=40 mai i 10x-2y=18 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$10x-10x-2y-20y=18-40$$

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.

$$-2y-20y=18-40$$

Tāpiri -2y ki te -20y.

$$-22y=18-40$$

Tāpiri 18 ki te -40.

$$-22y=-22$$

Whakawehea ngā taha e rua ki te -22.

$$y=1$$

Whakaurua te 1 mō y ki 2x+4y=8. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x+4=8$$

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

$$2x=4$$

Whakawehea ngā taha e rua ki te 2.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=1$$