Whakaoti i te {c}{40x+60y=480}{30x+15y=180}

$$\left. \begin{array} { c } { 40 x + 60 y = 480 } \\ { 30 x + 15 y = 180 } \end{array} \right.$$

Whakaoti mō x, y

$$x=3$$

$$y=6$$

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.

$$40x+60y=480,30x+15y=180$$

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

$$40x+60y=480$$

Me tango 60y mai i ngā taha e rua o te whārite.

$$40x=-60y+480$$

Whakawehea ngā taha e rua ki te 40.

$$x=\frac{1}{40}\left(-60y+480\right)$$

Whakareatia \frac{1}{40} ki te -60y+480.

$$x=-\frac{3}{2}y+12$$

Whakakapia te -\frac{3y}{2}+12 mō te x ki tērā atu whārite, 30x+15y=180.

$$30\left(-\frac{3}{2}y+12\right)+15y=180$$

Whakareatia 30 ki te -\frac{3y}{2}+12.

$$-45y+360+15y=180$$

Tāpiri -45y ki te 15y.

$$-30y+360=180$$

Me tango 360 mai i ngā taha e rua o te whārite.

$$-30y=-180$$

Whakawehea ngā taha e rua ki te -30.

$$y=6$$

Whakaurua te 6 mō y ki x=-\frac{3}{2}y+12. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{3}{2}\times 6+12$$

Whakareatia -\frac{3}{2} ki te 6.

$$x=-9+12$$

Tāpiri 12 ki te -9.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=6$$

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.

$$40x+60y=480,30x+15y=180$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}40&60\\30&15\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}480\\180\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}40&60\\30&15\end{matrix}\right).

$$inverse(\left(\begin{matrix}40&60\\30&15\end{matrix}\right))\left(\begin{matrix}40&60\\30&15\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}40&60\\30&15\end{matrix}\right))\left(\begin{matrix}480\\180\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}40&60\\30&15\end{matrix}\right))\left(\begin{matrix}480\\180\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}40&60\\30&15\end{matrix}\right))\left(\begin{matrix}480\\180\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{15}{40\times 15-60\times 30}&-\frac{60}{40\times 15-60\times 30}\\-\frac{30}{40\times 15-60\times 30}&\frac{40}{40\times 15-60\times 30}\end{matrix}\right)\left(\begin{matrix}480\\180\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{80}&\frac{1}{20}\\\frac{1}{40}&-\frac{1}{30}\end{matrix}\right)\left(\begin{matrix}480\\180\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{80}\times 480+\frac{1}{20}\times 180\\\frac{1}{40}\times 480-\frac{1}{30}\times 180\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3\\6\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=3,y=6$$

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.

$$40x+60y=480,30x+15y=180$$

Kia ōrite ai a 40x me 30x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 30 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 40.

$$30\times 40x+30\times 60y=30\times 480,40\times 30x+40\times 15y=40\times 180$$

Whakarūnātia.

$$1200x+1800y=14400,1200x+600y=7200$$

Me tango 1200x+600y=7200 mai i 1200x+1800y=14400 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$1200x-1200x+1800y-600y=14400-7200$$

Tāpiri 1200x ki te -1200x. Ka whakakore atu ngā kupu 1200x me -1200x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$1800y-600y=14400-7200$$

Tāpiri 1800y ki te -600y.

$$1200y=14400-7200$$

Tāpiri 14400 ki te -7200.

$$1200y=7200$$

Whakawehea ngā taha e rua ki te 1200.

$$y=6$$

Whakaurua te 6 mō y ki 30x+15y=180. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$30x+15\times 6=180$$

Whakareatia 15 ki te 6.

$$30x+90=180$$

Me tango 90 mai i ngā taha e rua o te whārite.

$$30x=90$$

Whakawehea ngā taha e rua ki te 30.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=6$$