Whakaoti i te {l}{78x+40y=1280}{120x+8y=2800}

$$\left\{ \begin{array} { l } { 78 x + 40 y = 1280 } \\ { 120 x + 8 y = 2800 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{2120}{87} = 24\frac{32}{87} \approx 24.367816092$$

$$y = -\frac{450}{29} = -15\frac{15}{29} \approx -15.517241379$$

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.

$$78x+40y=1280,120x+8y=2800$$

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

$$78x+40y=1280$$

Me tango 40y mai i ngā taha e rua o te whārite.

$$78x=-40y+1280$$

Whakawehea ngā taha e rua ki te 78.

$$x=\frac{1}{78}\left(-40y+1280\right)$$

Whakareatia \frac{1}{78} ki te -40y+1280.

$$x=-\frac{20}{39}y+\frac{640}{39}$$

Whakakapia te \frac{-20y+640}{39} mō te x ki tērā atu whārite, 120x+8y=2800.

$$120\left(-\frac{20}{39}y+\frac{640}{39}\right)+8y=2800$$

Whakareatia 120 ki te \frac{-20y+640}{39}.

$$-\frac{800}{13}y+\frac{25600}{13}+8y=2800$$

Tāpiri -\frac{800y}{13} ki te 8y.

$$-\frac{696}{13}y+\frac{25600}{13}=2800$$

Me tango \frac{25600}{13} mai i ngā taha e rua o te whārite.

$$-\frac{696}{13}y=\frac{10800}{13}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{696}{13}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-\frac{450}{29}$$

Whakaurua te -\frac{450}{29} mō y ki x=-\frac{20}{39}y+\frac{640}{39}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{20}{39}\left(-\frac{450}{29}\right)+\frac{640}{39}$$

Whakareatia -\frac{20}{39} ki te -\frac{450}{29} 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.

$$x=\frac{3000}{377}+\frac{640}{39}$$

Tāpiri \frac{640}{39} ki te \frac{3000}{377} 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.

$$x=\frac{2120}{87}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{2120}{87},y=-\frac{450}{29}$$

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.

$$78x+40y=1280,120x+8y=2800$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}78&40\\120&8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1280\\2800\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}78&40\\120&8\end{matrix}\right).

$$inverse(\left(\begin{matrix}78&40\\120&8\end{matrix}\right))\left(\begin{matrix}78&40\\120&8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}78&40\\120&8\end{matrix}\right))\left(\begin{matrix}1280\\2800\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}78&40\\120&8\end{matrix}\right))\left(\begin{matrix}1280\\2800\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}78&40\\120&8\end{matrix}\right))\left(\begin{matrix}1280\\2800\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{8}{78\times 8-40\times 120}&-\frac{40}{78\times 8-40\times 120}\\-\frac{120}{78\times 8-40\times 120}&\frac{78}{78\times 8-40\times 120}\end{matrix}\right)\left(\begin{matrix}1280\\2800\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{522}&\frac{5}{522}\\\frac{5}{174}&-\frac{13}{696}\end{matrix}\right)\left(\begin{matrix}1280\\2800\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{522}\times 1280+\frac{5}{522}\times 2800\\\frac{5}{174}\times 1280-\frac{13}{696}\times 2800\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2120}{87}\\-\frac{450}{29}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{2120}{87},y=-\frac{450}{29}$$

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.

$$78x+40y=1280,120x+8y=2800$$

Kia ōrite ai a 78x me 120x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 120 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 78.

$$120\times 78x+120\times 40y=120\times 1280,78\times 120x+78\times 8y=78\times 2800$$

Whakarūnātia.

$$9360x+4800y=153600,9360x+624y=218400$$

Me tango 9360x+624y=218400 mai i 9360x+4800y=153600 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$9360x-9360x+4800y-624y=153600-218400$$

Tāpiri 9360x ki te -9360x. Ka whakakore atu ngā kupu 9360x me -9360x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$4800y-624y=153600-218400$$

Tāpiri 4800y ki te -624y.

$$4176y=153600-218400$$

Tāpiri 153600 ki te -218400.

$$4176y=-64800$$

Whakawehea ngā taha e rua ki te 4176.

$$y=-\frac{450}{29}$$

Whakaurua te -\frac{450}{29} mō y ki 120x+8y=2800. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$120x+8\left(-\frac{450}{29}\right)=2800$$

Whakareatia 8 ki te -\frac{450}{29}.

$$120x-\frac{3600}{29}=2800$$

Me tāpiri \frac{3600}{29} ki ngā taha e rua o te whārite.

$$120x=\frac{84800}{29}$$

Whakawehea ngā taha e rua ki te 120.

$$x=\frac{2120}{87}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{2120}{87},y=-\frac{450}{29}$$