Whakaoti i te {r}{5x-2y=-1}{x+4y=35}

$$\left. \begin{array} { r } { 5 x - 2 y = - 1 } \\ { x + 4 y = 35 } \end{array} \right.$$

Whakaoti mō x, y

$$x=3$$

$$y=8$$

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.

$$5x-2y=-1,x+4y=35$$

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

$$5x-2y=-1$$

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

$$5x=2y-1$$

Whakawehea ngā taha e rua ki te 5.

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

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

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

Whakakapia te \frac{2y-1}{5} mō te x ki tērā atu whārite, x+4y=35.

$$\frac{2}{5}y-\frac{1}{5}+4y=35$$

Tāpiri \frac{2y}{5} ki te 4y.

$$\frac{22}{5}y-\frac{1}{5}=35$$

Me tāpiri \frac{1}{5} ki ngā taha e rua o te whārite.

$$\frac{22}{5}y=\frac{176}{5}$$

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.

$$y=8$$

Whakaurua te 8 mō y ki x=\frac{2}{5}y-\frac{1}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{2}{5}\times 8-\frac{1}{5}$$

Whakareatia \frac{2}{5} ki te 8.

$$x=\frac{16-1}{5}$$

Tāpiri -\frac{1}{5} ki te \frac{16}{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.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=8$$

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.

$$5x-2y=-1,x+4y=35$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&-2\\1&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-1\\35\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}5&-2\\1&4\end{matrix}\right))\left(\begin{matrix}5&-2\\1&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-2\\1&4\end{matrix}\right))\left(\begin{matrix}-1\\35\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-2\\1&4\end{matrix}\right))\left(\begin{matrix}-1\\35\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-2\\1&4\end{matrix}\right))\left(\begin{matrix}-1\\35\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{5\times 4-\left(-2\right)}&-\frac{-2}{5\times 4-\left(-2\right)}\\-\frac{1}{5\times 4-\left(-2\right)}&\frac{5}{5\times 4-\left(-2\right)}\end{matrix}\right)\left(\begin{matrix}-1\\35\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{11}&\frac{1}{11}\\-\frac{1}{22}&\frac{5}{22}\end{matrix}\right)\left(\begin{matrix}-1\\35\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{11}\left(-1\right)+\frac{1}{11}\times 35\\-\frac{1}{22}\left(-1\right)+\frac{5}{22}\times 35\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3\\8\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=3,y=8$$

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.

$$5x-2y=-1,x+4y=35$$

Kia ōrite ai a 5x me x, 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 5.

$$5x-2y=-1,5x+5\times 4y=5\times 35$$

Whakarūnātia.

$$5x-2y=-1,5x+20y=175$$

Me tango 5x+20y=175 mai i 5x-2y=-1 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$5x-5x-2y-20y=-1-175$$

Tāpiri 5x ki te -5x. Ka whakakore atu ngā kupu 5x me -5x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-2y-20y=-1-175$$

Tāpiri -2y ki te -20y.

$$-22y=-1-175$$

Tāpiri -1 ki te -175.

$$-22y=-176$$

Whakawehea ngā taha e rua ki te -22.

$$y=8$$

Whakaurua te 8 mō y ki x+4y=35. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x+4\times 8=35$$

Whakareatia 4 ki te 8.

$$x+32=35$$

Me tango 32 mai i ngā taha e rua o te whārite.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=8$$