Whakaoti i te {l}{6.5=60x+30y}{6=40y+50x}

$$\left\{ \begin{array} { l } { 6.5 = 60 x + 30 y } \\ { 6 = 40 y + 50 x } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{4}{45}\approx 0.088888889$$

$$y=\frac{7}{180}\approx 0.038888889$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$60x+30y=6.5$$

Whakaarohia te whārite tuarua. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$40y+50x=6$$

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.

$$60x+30y=6.5,50x+40y=6$$

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

$$60x+30y=6.5$$

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

$$60x=-30y+6.5$$

Whakawehea ngā taha e rua ki te 60.

$$x=\frac{1}{60}\left(-30y+6.5\right)$$

Whakareatia \frac{1}{60} ki te -30y+6.5.

$$x=-\frac{1}{2}y+\frac{13}{120}$$

Whakakapia te -\frac{y}{2}+\frac{13}{120} mō te x ki tērā atu whārite, 50x+40y=6.

$$50\left(-\frac{1}{2}y+\frac{13}{120}\right)+40y=6$$

Whakareatia 50 ki te -\frac{y}{2}+\frac{13}{120}.

$$-25y+\frac{65}{12}+40y=6$$

Tāpiri -25y ki te 40y.

$$15y+\frac{65}{12}=6$$

Me tango \frac{65}{12} mai i ngā taha e rua o te whārite.

$$15y=\frac{7}{12}$$

Whakawehea ngā taha e rua ki te 15.

$$y=\frac{7}{180}$$

Whakaurua te \frac{7}{180} mō y ki x=-\frac{1}{2}y+\frac{13}{120}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{2}\times \frac{7}{180}+\frac{13}{120}$$

Whakareatia -\frac{1}{2} ki te \frac{7}{180} 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.

$$x=-\frac{7}{360}+\frac{13}{120}$$

Tāpiri \frac{13}{120} ki te -\frac{7}{360} 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.

$$x=\frac{4}{45}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{4}{45},y=\frac{7}{180}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$60x+30y=6.5$$

Whakaarohia te whārite tuarua. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$40y+50x=6$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$60x+30y=6.5,50x+40y=6$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}60&30\\50&40\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}6.5\\6\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}60&30\\50&40\end{matrix}\right).

$$inverse(\left(\begin{matrix}60&30\\50&40\end{matrix}\right))\left(\begin{matrix}60&30\\50&40\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}60&30\\50&40\end{matrix}\right))\left(\begin{matrix}6.5\\6\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}60&30\\50&40\end{matrix}\right))\left(\begin{matrix}6.5\\6\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}60&30\\50&40\end{matrix}\right))\left(\begin{matrix}6.5\\6\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{40}{60\times 40-30\times 50}&-\frac{30}{60\times 40-30\times 50}\\-\frac{50}{60\times 40-30\times 50}&\frac{60}{60\times 40-30\times 50}\end{matrix}\right)\left(\begin{matrix}6.5\\6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{45}&-\frac{1}{30}\\-\frac{1}{18}&\frac{1}{15}\end{matrix}\right)\left(\begin{matrix}6.5\\6\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{45}\times 6.5-\frac{1}{30}\times 6\\-\frac{1}{18}\times 6.5+\frac{1}{15}\times 6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{45}\\\frac{7}{180}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{4}{45},y=\frac{7}{180}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$60x+30y=6.5$$

Whakaarohia te whārite tuarua. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$40y+50x=6$$

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.

$$60x+30y=6.5,50x+40y=6$$

Kia ōrite ai a 60x me 50x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 50 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 60.

$$50\times 60x+50\times 30y=50\times 6.5,60\times 50x+60\times 40y=60\times 6$$

Whakarūnātia.

$$3000x+1500y=325,3000x+2400y=360$$

Me tango 3000x+2400y=360 mai i 3000x+1500y=325 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3000x-3000x+1500y-2400y=325-360$$

Tāpiri 3000x ki te -3000x. Ka whakakore atu ngā kupu 3000x me -3000x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$1500y-2400y=325-360$$

Tāpiri 1500y ki te -2400y.

$$-900y=325-360$$

Tāpiri 325 ki te -360.

$$-900y=-35$$

Whakawehea ngā taha e rua ki te -900.

$$y=\frac{7}{180}$$

Whakaurua te \frac{7}{180} mō y ki 50x+40y=6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$50x+40\times \frac{7}{180}=6$$

Whakareatia 40 ki te \frac{7}{180}.

$$50x+\frac{14}{9}=6$$

Me tango \frac{14}{9} mai i ngā taha e rua o te whārite.

$$50x=\frac{40}{9}$$

Whakawehea ngā taha e rua ki te 50.

$$x=\frac{4}{45}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{4}{45},y=\frac{7}{180}$$