Whakaoti i te {l}{40x+30y=500}{60x+15y=600}

$$\left. \begin{array} { l } { 40 x + 30 y = 500 } \\ { 60 x + 15 y = 600 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{35}{4} = 8\frac{3}{4} = 8.75$$

$$y=5$$

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.

$$40x+30y=500,60x+15y=600$$

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

$$40x+30y=500$$

Me tango 30y mai i ngā taha e rua o te whārite.

$$40x=-30y+500$$

Whakawehea ngā taha e rua ki te 40.

$$x=\frac{1}{40}\left(-30y+500\right)$$

Whakareatia \frac{1}{40} ki te -30y+500.

$$x=-\frac{3}{4}y+\frac{25}{2}$$

Whakakapia te -\frac{3y}{4}+\frac{25}{2} mō te x ki tērā atu whārite, 60x+15y=600.

$$60\left(-\frac{3}{4}y+\frac{25}{2}\right)+15y=600$$

Whakareatia 60 ki te -\frac{3y}{4}+\frac{25}{2}.

$$-45y+750+15y=600$$

Tāpiri -45y ki te 15y.

$$-30y+750=600$$

Me tango 750 mai i ngā taha e rua o te whārite.

$$-30y=-150$$

Whakawehea ngā taha e rua ki te -30.

$$y=5$$

Whakaurua te 5 mō y ki x=-\frac{3}{4}y+\frac{25}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{3}{4}\times 5+\frac{25}{2}$$

Whakareatia -\frac{3}{4} ki te 5.

$$x=-\frac{15}{4}+\frac{25}{2}$$

Tāpiri \frac{25}{2} ki te -\frac{15}{4} 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.

$$x=\frac{35}{4}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{35}{4},y=5$$

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.

$$40x+30y=500,60x+15y=600$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}40&30\\60&15\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}500\\600\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}40&30\\60&15\end{matrix}\right))\left(\begin{matrix}40&30\\60&15\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}40&30\\60&15\end{matrix}\right))\left(\begin{matrix}500\\600\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}40&30\\60&15\end{matrix}\right))\left(\begin{matrix}500\\600\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}40&30\\60&15\end{matrix}\right))\left(\begin{matrix}500\\600\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{15}{40\times 15-30\times 60}&-\frac{30}{40\times 15-30\times 60}\\-\frac{60}{40\times 15-30\times 60}&\frac{40}{40\times 15-30\times 60}\end{matrix}\right)\left(\begin{matrix}500\\600\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{80}&\frac{1}{40}\\\frac{1}{20}&-\frac{1}{30}\end{matrix}\right)\left(\begin{matrix}500\\600\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{80}\times 500+\frac{1}{40}\times 600\\\frac{1}{20}\times 500-\frac{1}{30}\times 600\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{35}{4}\\5\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{35}{4},y=5$$

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.

$$40x+30y=500,60x+15y=600$$

Kia ōrite ai a 40x me 60x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 60 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 40.

$$60\times 40x+60\times 30y=60\times 500,40\times 60x+40\times 15y=40\times 600$$

Whakarūnātia.

$$2400x+1800y=30000,2400x+600y=24000$$

Me tango 2400x+600y=24000 mai i 2400x+1800y=30000 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2400x-2400x+1800y-600y=30000-24000$$

Tāpiri 2400x ki te -2400x. Ka whakakore atu ngā kupu 2400x me -2400x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$1800y-600y=30000-24000$$

Tāpiri 1800y ki te -600y.

$$1200y=30000-24000$$

Tāpiri 30000 ki te -24000.

$$1200y=6000$$

Whakawehea ngā taha e rua ki te 1200.

$$y=5$$

Whakaurua te 5 mō y ki 60x+15y=600. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$60x+15\times 5=600$$

Whakareatia 15 ki te 5.

$$60x+75=600$$

Me tango 75 mai i ngā taha e rua o te whārite.

$$60x=525$$

Whakawehea ngā taha e rua ki te 60.

$$x=\frac{35}{4}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{35}{4},y=5$$