Whakaoti i te {l}{x+y=30}{2x+25y=698}

$$\left\{ \begin{array} { l } { x + y = 30 } \\ { 2 x + 25 y = 698 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{52}{23} = 2\frac{6}{23} \approx 2.260869565$$

$$y = \frac{638}{23} = 27\frac{17}{23} \approx 27.739130435$$

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.

$$x+y=30,2x+25y=698$$

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

$$x+y=30$$

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

$$x=-y+30$$

Whakakapia te -y+30 mō te x ki tērā atu whārite, 2x+25y=698.

$$2\left(-y+30\right)+25y=698$$

Whakareatia 2 ki te -y+30.

$$-2y+60+25y=698$$

Tāpiri -2y ki te 25y.

$$23y+60=698$$

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

$$23y=638$$

Whakawehea ngā taha e rua ki te 23.

$$y=\frac{638}{23}$$

Whakaurua te \frac{638}{23} mō y ki x=-y+30. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{638}{23}+30$$

Tāpiri 30 ki te -\frac{638}{23}.

$$x=\frac{52}{23}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{52}{23},y=\frac{638}{23}$$

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.

$$x+y=30,2x+25y=698$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\2&25\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}30\\698\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&1\\2&25\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&1\\2&25\end{matrix}\right))\left(\begin{matrix}1&1\\2&25\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\2&25\end{matrix}\right))\left(\begin{matrix}30\\698\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\2&25\end{matrix}\right))\left(\begin{matrix}30\\698\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\2&25\end{matrix}\right))\left(\begin{matrix}30\\698\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{25}{25-2}&-\frac{1}{25-2}\\-\frac{2}{25-2}&\frac{1}{25-2}\end{matrix}\right)\left(\begin{matrix}30\\698\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{25}{23}&-\frac{1}{23}\\-\frac{2}{23}&\frac{1}{23}\end{matrix}\right)\left(\begin{matrix}30\\698\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{25}{23}\times 30-\frac{1}{23}\times 698\\-\frac{2}{23}\times 30+\frac{1}{23}\times 698\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{52}{23}\\\frac{638}{23}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{52}{23},y=\frac{638}{23}$$

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.

$$x+y=30,2x+25y=698$$

Kia ōrite ai a x me 2x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$2x+2y=2\times 30,2x+25y=698$$

Whakarūnātia.

$$2x+2y=60,2x+25y=698$$

Me tango 2x+25y=698 mai i 2x+2y=60 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2x-2x+2y-25y=60-698$$

Tāpiri 2x ki te -2x. Ka whakakore atu ngā kupu 2x me -2x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$2y-25y=60-698$$

Tāpiri 2y ki te -25y.

$$-23y=60-698$$

Tāpiri 60 ki te -698.

$$-23y=-638$$

Whakawehea ngā taha e rua ki te -23.

$$y=\frac{638}{23}$$

Whakaurua te \frac{638}{23} mō y ki 2x+25y=698. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x+25\times \frac{638}{23}=698$$

Whakareatia 25 ki te \frac{638}{23}.

$$2x+\frac{15950}{23}=698$$

Me tango \frac{15950}{23} mai i ngā taha e rua o te whārite.

$$2x=\frac{104}{23}$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{52}{23}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{52}{23},y=\frac{638}{23}$$