Whakaoti i te {l}{48x+40y=1280}{120x+80y=2800}

$$\left\{ \begin{array} { l } { 48 x + 40 y = 1280 } \\ { 120 x + 80 y = 2800 } \end{array} \right.$$

Whakaoti mō x, y

$$x=10$$

$$y=20$$

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.

$$48x+40y=1280,120x+80y=2800$$

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

$$48x+40y=1280$$

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

$$48x=-40y+1280$$

Whakawehea ngā taha e rua ki te 48.

$$x=\frac{1}{48}\left(-40y+1280\right)$$

Whakareatia \frac{1}{48} ki te -40y+1280.

$$x=-\frac{5}{6}y+\frac{80}{3}$$

Whakakapia te -\frac{5y}{6}+\frac{80}{3} mō te x ki tērā atu whārite, 120x+80y=2800.

$$120\left(-\frac{5}{6}y+\frac{80}{3}\right)+80y=2800$$

Whakareatia 120 ki te -\frac{5y}{6}+\frac{80}{3}.

$$-100y+3200+80y=2800$$

Tāpiri -100y ki te 80y.

$$-20y+3200=2800$$

Me tango 3200 mai i ngā taha e rua o te whārite.

$$-20y=-400$$

Whakawehea ngā taha e rua ki te -20.

$$y=20$$

Whakaurua te 20 mō y ki x=-\frac{5}{6}y+\frac{80}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{5}{6}\times 20+\frac{80}{3}$$

Whakareatia -\frac{5}{6} ki te 20.

$$x=\frac{-50+80}{3}$$

Tāpiri \frac{80}{3} ki te -\frac{50}{3} 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.

$$x=10$$

Kua oti te pūnaha te whakatau.

$$x=10,y=20$$

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.

$$48x+40y=1280,120x+80y=2800$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}48&40\\120&80\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1280\\2800\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}48&40\\120&80\end{matrix}\right).

$$inverse(\left(\begin{matrix}48&40\\120&80\end{matrix}\right))\left(\begin{matrix}48&40\\120&80\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}48&40\\120&80\end{matrix}\right))\left(\begin{matrix}1280\\2800\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}48&40\\120&80\end{matrix}\right))\left(\begin{matrix}1280\\2800\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}48&40\\120&80\end{matrix}\right))\left(\begin{matrix}1280\\2800\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{80}{48\times 80-40\times 120}&-\frac{40}{48\times 80-40\times 120}\\-\frac{120}{48\times 80-40\times 120}&\frac{48}{48\times 80-40\times 120}\end{matrix}\right)\left(\begin{matrix}1280\\2800\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{12}&\frac{1}{24}\\\frac{1}{8}&-\frac{1}{20}\end{matrix}\right)\left(\begin{matrix}1280\\2800\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{12}\times 1280+\frac{1}{24}\times 2800\\\frac{1}{8}\times 1280-\frac{1}{20}\times 2800\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}10\\20\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=10,y=20$$

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.

$$48x+40y=1280,120x+80y=2800$$

Kia ōrite ai a 48x 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 48.

$$120\times 48x+120\times 40y=120\times 1280,48\times 120x+48\times 80y=48\times 2800$$

Whakarūnātia.

$$5760x+4800y=153600,5760x+3840y=134400$$

Me tango 5760x+3840y=134400 mai i 5760x+4800y=153600 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$5760x-5760x+4800y-3840y=153600-134400$$

Tāpiri 5760x ki te -5760x. Ka whakakore atu ngā kupu 5760x me -5760x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$4800y-3840y=153600-134400$$

Tāpiri 4800y ki te -3840y.

$$960y=153600-134400$$

Tāpiri 153600 ki te -134400.

$$960y=19200$$

Whakawehea ngā taha e rua ki te 960.

$$y=20$$

Whakaurua te 20 mō y ki 120x+80y=2800. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$120x+80\times 20=2800$$

Whakareatia 80 ki te 20.

$$120x+1600=2800$$

Me tango 1600 mai i ngā taha e rua o te whārite.

$$120x=1200$$

Whakawehea ngā taha e rua ki te 120.

$$x=10$$

Kua oti te pūnaha te whakatau.

$$x=10,y=20$$