Whakaoti i te {l}{11x+19y=25}{19x+11y=15}

$$\left\{ \begin{array} { l } { 11 x + 19 y = 25 } \\ { 19 x + 11 y = 15 } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{1}{24}\approx 0.041666667$$

$$y = \frac{31}{24} = 1\frac{7}{24} \approx 1.291666667$$

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.

$$11x+19y=25,19x+11y=15$$

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

$$11x+19y=25$$

Me tango 19y mai i ngā taha e rua o te whārite.

$$11x=-19y+25$$

Whakawehea ngā taha e rua ki te 11.

$$x=\frac{1}{11}\left(-19y+25\right)$$

Whakareatia \frac{1}{11} ki te -19y+25.

$$x=-\frac{19}{11}y+\frac{25}{11}$$

Whakakapia te \frac{-19y+25}{11} mō te x ki tērā atu whārite, 19x+11y=15.

$$19\left(-\frac{19}{11}y+\frac{25}{11}\right)+11y=15$$

Whakareatia 19 ki te \frac{-19y+25}{11}.

$$-\frac{361}{11}y+\frac{475}{11}+11y=15$$

Tāpiri -\frac{361y}{11} ki te 11y.

$$-\frac{240}{11}y+\frac{475}{11}=15$$

Me tango \frac{475}{11} mai i ngā taha e rua o te whārite.

$$-\frac{240}{11}y=-\frac{310}{11}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{240}{11}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{31}{24}$$

Whakaurua te \frac{31}{24} mō y ki x=-\frac{19}{11}y+\frac{25}{11}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{19}{11}\times \frac{31}{24}+\frac{25}{11}$$

Whakareatia -\frac{19}{11} ki te \frac{31}{24} 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.

$$x=-\frac{589}{264}+\frac{25}{11}$$

Tāpiri \frac{25}{11} ki te -\frac{589}{264} 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.

$$x=\frac{1}{24}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{1}{24},y=\frac{31}{24}$$

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.

$$11x+19y=25,19x+11y=15$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}11&19\\19&11\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}25\\15\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}11&19\\19&11\end{matrix}\right).

$$inverse(\left(\begin{matrix}11&19\\19&11\end{matrix}\right))\left(\begin{matrix}11&19\\19&11\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}11&19\\19&11\end{matrix}\right))\left(\begin{matrix}25\\15\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}11&19\\19&11\end{matrix}\right))\left(\begin{matrix}25\\15\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}11&19\\19&11\end{matrix}\right))\left(\begin{matrix}25\\15\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{11}{11\times 11-19\times 19}&-\frac{19}{11\times 11-19\times 19}\\-\frac{19}{11\times 11-19\times 19}&\frac{11}{11\times 11-19\times 19}\end{matrix}\right)\left(\begin{matrix}25\\15\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{11}{240}&\frac{19}{240}\\\frac{19}{240}&-\frac{11}{240}\end{matrix}\right)\left(\begin{matrix}25\\15\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{11}{240}\times 25+\frac{19}{240}\times 15\\\frac{19}{240}\times 25-\frac{11}{240}\times 15\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{24}\\\frac{31}{24}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{1}{24},y=\frac{31}{24}$$

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.

$$11x+19y=25,19x+11y=15$$

Kia ōrite ai a 11x me 19x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 19 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 11.

$$19\times 11x+19\times 19y=19\times 25,11\times 19x+11\times 11y=11\times 15$$

Whakarūnātia.

$$209x+361y=475,209x+121y=165$$

Me tango 209x+121y=165 mai i 209x+361y=475 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$209x-209x+361y-121y=475-165$$

Tāpiri 209x ki te -209x. Ka whakakore atu ngā kupu 209x me -209x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$361y-121y=475-165$$

Tāpiri 361y ki te -121y.

$$240y=475-165$$

Tāpiri 475 ki te -165.

$$240y=310$$

Whakawehea ngā taha e rua ki te 240.

$$y=\frac{31}{24}$$

Whakaurua te \frac{31}{24} mō y ki 19x+11y=15. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$19x+11\times \frac{31}{24}=15$$

Whakareatia 11 ki te \frac{31}{24}.

$$19x+\frac{341}{24}=15$$

Me tango \frac{341}{24} mai i ngā taha e rua o te whārite.

$$19x=\frac{19}{24}$$

Whakawehea ngā taha e rua ki te 19.

$$x=\frac{1}{24}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{1}{24},y=\frac{31}{24}$$