Whakaoti i te {l}{15x+17y=21-1/11}{17x+15y=11-11}

$$\left. \begin{array} { l } { 15 x + 17 y = 21 - \frac { 1 } { 11 } } \\ { 17 x + 15 y = 11 - 11 } \end{array} \right.$$

Whakaoti mō x, y

$$x = -\frac{1725}{352} = -4\frac{317}{352} \approx -4.900568182$$

$$y = \frac{1955}{352} = 5\frac{195}{352} \approx 5.553977273$$

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.

$$15x+17y=\frac{230}{11},17x+15y=0$$

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

$$15x+17y=\frac{230}{11}$$

Me tango 17y mai i ngā taha e rua o te whārite.

$$15x=-17y+\frac{230}{11}$$

Whakawehea ngā taha e rua ki te 15.

$$x=\frac{1}{15}\left(-17y+\frac{230}{11}\right)$$

Whakareatia \frac{1}{15} ki te -17y+\frac{230}{11}.

$$x=-\frac{17}{15}y+\frac{46}{33}$$

Whakakapia te -\frac{17y}{15}+\frac{46}{33} mō te x ki tērā atu whārite, 17x+15y=0.

$$17\left(-\frac{17}{15}y+\frac{46}{33}\right)+15y=0$$

Whakareatia 17 ki te -\frac{17y}{15}+\frac{46}{33}.

$$-\frac{289}{15}y+\frac{782}{33}+15y=0$$

Tāpiri -\frac{289y}{15} ki te 15y.

$$-\frac{64}{15}y+\frac{782}{33}=0$$

Me tango \frac{782}{33} mai i ngā taha e rua o te whārite.

$$-\frac{64}{15}y=-\frac{782}{33}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{64}{15}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{1955}{352}$$

Whakaurua te \frac{1955}{352} mō y ki x=-\frac{17}{15}y+\frac{46}{33}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{17}{15}\times \frac{1955}{352}+\frac{46}{33}$$

Whakareatia -\frac{17}{15} ki te \frac{1955}{352} 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.

$$x=-\frac{6647}{1056}+\frac{46}{33}$$

Tāpiri \frac{46}{33} ki te -\frac{6647}{1056} 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.

$$x=-\frac{1725}{352}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{1725}{352},y=\frac{1955}{352}$$

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.

$$15x+17y=\frac{230}{11},17x+15y=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}15&17\\17&15\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{230}{11}\\0\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}15&17\\17&15\end{matrix}\right).

$$inverse(\left(\begin{matrix}15&17\\17&15\end{matrix}\right))\left(\begin{matrix}15&17\\17&15\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}15&17\\17&15\end{matrix}\right))\left(\begin{matrix}\frac{230}{11}\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}15&17\\17&15\end{matrix}\right))\left(\begin{matrix}\frac{230}{11}\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}15&17\\17&15\end{matrix}\right))\left(\begin{matrix}\frac{230}{11}\\0\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{15}{15\times 15-17\times 17}&-\frac{17}{15\times 15-17\times 17}\\-\frac{17}{15\times 15-17\times 17}&\frac{15}{15\times 15-17\times 17}\end{matrix}\right)\left(\begin{matrix}\frac{230}{11}\\0\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{15}{64}&\frac{17}{64}\\\frac{17}{64}&-\frac{15}{64}\end{matrix}\right)\left(\begin{matrix}\frac{230}{11}\\0\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{15}{64}\times \frac{230}{11}\\\frac{17}{64}\times \frac{230}{11}\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1725}{352}\\\frac{1955}{352}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{1725}{352},y=\frac{1955}{352}$$

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.

$$15x+17y=\frac{230}{11},17x+15y=0$$

Kia ōrite ai a 15x me 17x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 17 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 15.

$$17\times 15x+17\times 17y=17\times \frac{230}{11},15\times 17x+15\times 15y=0$$

Whakarūnātia.

$$255x+289y=\frac{3910}{11},255x+225y=0$$

Me tango 255x+225y=0 mai i 255x+289y=\frac{3910}{11} mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$255x-255x+289y-225y=\frac{3910}{11}$$

Tāpiri 255x ki te -255x. Ka whakakore atu ngā kupu 255x me -255x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$289y-225y=\frac{3910}{11}$$

Tāpiri 289y ki te -225y.

$$64y=\frac{3910}{11}$$

Whakawehea ngā taha e rua ki te 64.

$$y=\frac{1955}{352}$$

Whakaurua te \frac{1955}{352} mō y ki 17x+15y=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$17x+15\times \frac{1955}{352}=0$$

Whakareatia 15 ki te \frac{1955}{352}.

$$17x+\frac{29325}{352}=0$$

Me tango \frac{29325}{352} mai i ngā taha e rua o te whārite.

$$17x=-\frac{29325}{352}$$

Whakawehea ngā taha e rua ki te 17.

$$x=-\frac{1725}{352}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{1725}{352},y=\frac{1955}{352}$$