Whakaoti i te {l}{x+y=250}{x/19+y/10=19}

$$\left. \begin{array} { l } { x + y = 250 } \\ { \frac { x } { 19 } + \frac { y } { 10 } = 19 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{380}{3} = 126\frac{2}{3} \approx 126.666666667$$

$$y = \frac{370}{3} = 123\frac{1}{3} \approx 123.333333333$$

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.

$$x+y=250,\frac{1}{19}x+\frac{1}{10}y=19$$

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

$$x+y=250$$

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

$$x=-y+250$$

Whakakapia te -y+250 mō te x ki tērā atu whārite, \frac{1}{19}x+\frac{1}{10}y=19.

$$\frac{1}{19}\left(-y+250\right)+\frac{1}{10}y=19$$

Whakareatia \frac{1}{19} ki te -y+250.

$$-\frac{1}{19}y+\frac{250}{19}+\frac{1}{10}y=19$$

Tāpiri -\frac{y}{19} ki te \frac{y}{10}.

$$\frac{9}{190}y+\frac{250}{19}=19$$

Me tango \frac{250}{19} mai i ngā taha e rua o te whārite.

$$\frac{9}{190}y=\frac{111}{19}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{9}{190}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{370}{3}$$

Whakaurua te \frac{370}{3} mō y ki x=-y+250. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{370}{3}+250$$

Tāpiri 250 ki te -\frac{370}{3}.

$$x=\frac{380}{3}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{380}{3},y=\frac{370}{3}$$

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.

$$x+y=250,\frac{1}{19}x+\frac{1}{10}y=19$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\\frac{1}{19}&\frac{1}{10}\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}250\\19\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&1\\\frac{1}{19}&\frac{1}{10}\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&1\\\frac{1}{19}&\frac{1}{10}\end{matrix}\right))\left(\begin{matrix}1&1\\\frac{1}{19}&\frac{1}{10}\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\\frac{1}{19}&\frac{1}{10}\end{matrix}\right))\left(\begin{matrix}250\\19\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\\frac{1}{19}&\frac{1}{10}\end{matrix}\right))\left(\begin{matrix}250\\19\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\\frac{1}{19}&\frac{1}{10}\end{matrix}\right))\left(\begin{matrix}250\\19\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te poukapa kōaro ko \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), kia tuhia anō ai te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{\frac{1}{10}}{\frac{1}{10}-\frac{1}{19}}&-\frac{1}{\frac{1}{10}-\frac{1}{19}}\\-\frac{\frac{1}{19}}{\frac{1}{10}-\frac{1}{19}}&\frac{1}{\frac{1}{10}-\frac{1}{19}}\end{matrix}\right)\left(\begin{matrix}250\\19\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{19}{9}&-\frac{190}{9}\\-\frac{10}{9}&\frac{190}{9}\end{matrix}\right)\left(\begin{matrix}250\\19\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{19}{9}\times 250-\frac{190}{9}\times 19\\-\frac{10}{9}\times 250+\frac{190}{9}\times 19\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{380}{3}\\\frac{370}{3}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{380}{3},y=\frac{370}{3}$$

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.

$$x+y=250,\frac{1}{19}x+\frac{1}{10}y=19$$

Kia ōrite ai a x me \frac{x}{19}, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te \frac{1}{19} me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$\frac{1}{19}x+\frac{1}{19}y=\frac{1}{19}\times 250,\frac{1}{19}x+\frac{1}{10}y=19$$

Whakarūnātia.

$$\frac{1}{19}x+\frac{1}{19}y=\frac{250}{19},\frac{1}{19}x+\frac{1}{10}y=19$$

Me tango \frac{1}{19}x+\frac{1}{10}y=19 mai i \frac{1}{19}x+\frac{1}{19}y=\frac{250}{19} mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$\frac{1}{19}x-\frac{1}{19}x+\frac{1}{19}y-\frac{1}{10}y=\frac{250}{19}-19$$

Tāpiri \frac{x}{19} ki te -\frac{x}{19}. Ka whakakore atu ngā kupu \frac{x}{19} me -\frac{x}{19}, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$\frac{1}{19}y-\frac{1}{10}y=\frac{250}{19}-19$$

Tāpiri \frac{y}{19} ki te -\frac{y}{10}.

$$-\frac{9}{190}y=\frac{250}{19}-19$$

Tāpiri \frac{250}{19} ki te -19.

$$-\frac{9}{190}y=-\frac{111}{19}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{9}{190}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{370}{3}$$

Whakaurua te \frac{370}{3} mō y ki \frac{1}{19}x+\frac{1}{10}y=19. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$\frac{1}{19}x+\frac{1}{10}\times \frac{370}{3}=19$$

Whakareatia \frac{1}{10} ki te \frac{370}{3} 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.

$$\frac{1}{19}x+\frac{37}{3}=19$$

Me tango \frac{37}{3} mai i ngā taha e rua o te whārite.

$$\frac{1}{19}x=\frac{20}{3}$$

Me whakarea ngā taha e rua ki te 19.

$$x=\frac{380}{3}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{380}{3},y=\frac{370}{3}$$