Whakaoti i te {l}{x/60+y/30=6*5}{x/50+y/40=6}

$$\left\{ \begin{array} { l } { \frac { x } { 60 } + \frac { y } { 30 } = 6 \cdot 5 } \\ { \frac { x } { 50 } + \frac { y } { 40 } = 6 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-2200$$

$$y=2000$$

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.

$$\frac{1}{60}x+\frac{1}{30}y=30,\frac{1}{50}x+\frac{1}{40}y=6$$

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

$$\frac{1}{60}x+\frac{1}{30}y=30$$

Me tango \frac{y}{30} mai i ngā taha e rua o te whārite.

$$\frac{1}{60}x=-\frac{1}{30}y+30$$

Me whakarea ngā taha e rua ki te 60.

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

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

$$x=-2y+1800$$

Whakakapia te -2y+1800 mō te x ki tērā atu whārite, \frac{1}{50}x+\frac{1}{40}y=6.

$$\frac{1}{50}\left(-2y+1800\right)+\frac{1}{40}y=6$$

Whakareatia \frac{1}{50} ki te -2y+1800.

$$-\frac{1}{25}y+36+\frac{1}{40}y=6$$

Tāpiri -\frac{y}{25} ki te \frac{y}{40}.

$$-\frac{3}{200}y+36=6$$

Me tango 36 mai i ngā taha e rua o te whārite.

$$-\frac{3}{200}y=-30$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{3}{200}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=2000$$

Whakaurua te 2000 mō y ki x=-2y+1800. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-2\times 2000+1800$$

Whakareatia -2 ki te 2000.

$$x=-4000+1800$$

Tāpiri 1800 ki te -4000.

$$x=-2200$$

Kua oti te pūnaha te whakatau.

$$x=-2200,y=2000$$

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.

$$\frac{1}{60}x+\frac{1}{30}y=30,\frac{1}{50}x+\frac{1}{40}y=6$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}\frac{1}{60}&\frac{1}{30}\\\frac{1}{50}&\frac{1}{40}\end{matrix}\right).

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

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

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

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

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

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{\frac{1}{40}}{\frac{1}{60}\times \frac{1}{40}-\frac{1}{30}\times \frac{1}{50}}&-\frac{\frac{1}{30}}{\frac{1}{60}\times \frac{1}{40}-\frac{1}{30}\times \frac{1}{50}}\\-\frac{\frac{1}{50}}{\frac{1}{60}\times \frac{1}{40}-\frac{1}{30}\times \frac{1}{50}}&\frac{\frac{1}{60}}{\frac{1}{60}\times \frac{1}{40}-\frac{1}{30}\times \frac{1}{50}}\end{matrix}\right)\left(\begin{matrix}30\\6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-100&\frac{400}{3}\\80&-\frac{200}{3}\end{matrix}\right)\left(\begin{matrix}30\\6\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-100\times 30+\frac{400}{3}\times 6\\80\times 30-\frac{200}{3}\times 6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-2200\\2000\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-2200,y=2000$$

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.

$$\frac{1}{60}x+\frac{1}{30}y=30,\frac{1}{50}x+\frac{1}{40}y=6$$

Kia ōrite ai a \frac{x}{60} me \frac{x}{50}, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te \frac{1}{50} me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te \frac{1}{60}.

$$\frac{1}{50}\times \frac{1}{60}x+\frac{1}{50}\times \frac{1}{30}y=\frac{1}{50}\times 30,\frac{1}{60}\times \frac{1}{50}x+\frac{1}{60}\times \frac{1}{40}y=\frac{1}{60}\times 6$$

Whakarūnātia.

$$\frac{1}{3000}x+\frac{1}{1500}y=\frac{3}{5},\frac{1}{3000}x+\frac{1}{2400}y=\frac{1}{10}$$

Me tango \frac{1}{3000}x+\frac{1}{2400}y=\frac{1}{10} mai i \frac{1}{3000}x+\frac{1}{1500}y=\frac{3}{5} mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$\frac{1}{3000}x-\frac{1}{3000}x+\frac{1}{1500}y-\frac{1}{2400}y=\frac{3}{5}-\frac{1}{10}$$

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

$$\frac{1}{1500}y-\frac{1}{2400}y=\frac{3}{5}-\frac{1}{10}$$

Tāpiri \frac{y}{1500} ki te -\frac{y}{2400}.

$$\frac{1}{4000}y=\frac{3}{5}-\frac{1}{10}$$

Tāpiri \frac{3}{5} ki te -\frac{1}{10} 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.

$$\frac{1}{4000}y=\frac{1}{2}$$

Me whakarea ngā taha e rua ki te 4000.

$$y=2000$$

Whakaurua te 2000 mō y ki \frac{1}{50}x+\frac{1}{40}y=6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$\frac{1}{50}x+\frac{1}{40}\times 2000=6$$

Whakareatia \frac{1}{40} ki te 2000.

$$\frac{1}{50}x+50=6$$

Me tango 50 mai i ngā taha e rua o te whārite.

$$\frac{1}{50}x=-44$$

Me whakarea ngā taha e rua ki te 50.

$$x=-2200$$

Kua oti te pūnaha te whakatau.

$$x=-2200,y=2000$$