Whakaoti i te {l}{25x+16y=72}{-5x+4y=0}

$$\left. \begin{array} { l } { 25 x + 16 y = 72 } \\ { - 5 x + 4 y = 0 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{8}{5} = 1\frac{3}{5} = 1.6$$

$$y=2$$

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.

$$25x+16y=72,-5x+4y=0$$

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

$$25x+16y=72$$

Me tango 16y mai i ngā taha e rua o te whārite.

$$25x=-16y+72$$

Whakawehea ngā taha e rua ki te 25.

$$x=\frac{1}{25}\left(-16y+72\right)$$

Whakareatia \frac{1}{25} ki te -16y+72.

$$x=-\frac{16}{25}y+\frac{72}{25}$$

Whakakapia te \frac{-16y+72}{25} mō te x ki tērā atu whārite, -5x+4y=0.

$$-5\left(-\frac{16}{25}y+\frac{72}{25}\right)+4y=0$$

Whakareatia -5 ki te \frac{-16y+72}{25}.

$$\frac{16}{5}y-\frac{72}{5}+4y=0$$

Tāpiri \frac{16y}{5} ki te 4y.

$$\frac{36}{5}y-\frac{72}{5}=0$$

Me tāpiri \frac{72}{5} ki ngā taha e rua o te whārite.

$$\frac{36}{5}y=\frac{72}{5}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{36}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=2$$

Whakaurua te 2 mō y ki x=-\frac{16}{25}y+\frac{72}{25}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{16}{25}\times 2+\frac{72}{25}$$

Whakareatia -\frac{16}{25} ki te 2.

$$x=\frac{-32+72}{25}$$

Tāpiri \frac{72}{25} ki te -\frac{32}{25} 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.

$$x=\frac{8}{5}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{8}{5},y=2$$

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.

$$25x+16y=72,-5x+4y=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}25&16\\-5&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}72\\0\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}25&16\\-5&4\end{matrix}\right).

$$inverse(\left(\begin{matrix}25&16\\-5&4\end{matrix}\right))\left(\begin{matrix}25&16\\-5&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}25&16\\-5&4\end{matrix}\right))\left(\begin{matrix}72\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}25&16\\-5&4\end{matrix}\right))\left(\begin{matrix}72\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}25&16\\-5&4\end{matrix}\right))\left(\begin{matrix}72\\0\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{25\times 4-16\left(-5\right)}&-\frac{16}{25\times 4-16\left(-5\right)}\\-\frac{-5}{25\times 4-16\left(-5\right)}&\frac{25}{25\times 4-16\left(-5\right)}\end{matrix}\right)\left(\begin{matrix}72\\0\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{45}&-\frac{4}{45}\\\frac{1}{36}&\frac{5}{36}\end{matrix}\right)\left(\begin{matrix}72\\0\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{45}\times 72\\\frac{1}{36}\times 72\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{8}{5}\\2\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{8}{5},y=2$$

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.

$$25x+16y=72,-5x+4y=0$$

Kia ōrite ai a 25x me -5x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -5 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 25.

$$-5\times 25x-5\times 16y=-5\times 72,25\left(-5\right)x+25\times 4y=0$$

Whakarūnātia.

$$-125x-80y=-360,-125x+100y=0$$

Me tango -125x+100y=0 mai i -125x-80y=-360 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-125x+125x-80y-100y=-360$$

Tāpiri -125x ki te 125x. Ka whakakore atu ngā kupu -125x me 125x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-80y-100y=-360$$

Tāpiri -80y ki te -100y.

$$-180y=-360$$

Whakawehea ngā taha e rua ki te -180.

$$y=2$$

Whakaurua te 2 mō y ki -5x+4y=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-5x+4\times 2=0$$

Whakareatia 4 ki te 2.

$$-5x+8=0$$

Me tango 8 mai i ngā taha e rua o te whārite.

$$-5x=-8$$

Whakawehea ngā taha e rua ki te -5.

$$x=\frac{8}{5}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{8}{5},y=2$$