Whakaoti i te {l}{x/3+y/4=-7/12}{x/2+frac{y}{3}=-frac{1}{6}}

$$\left\{ \begin{array} { l } { \frac { x } { 3 } + \frac { y } { 4 } = - \frac { 7 } { 12 } } \\ { \frac { x } { 2 } + \frac { y } { 3 } = - \frac { 1 } { 6 } } \end{array} \right.$$

Whakaoti mō x, y

$$x=11$$

$$y=-17$$

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.

$$\frac{1}{3}x+\frac{1}{4}y=-\frac{7}{12},\frac{1}{2}x+\frac{1}{3}y=-\frac{1}{6}$$

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

$$\frac{1}{3}x+\frac{1}{4}y=-\frac{7}{12}$$

Me tango \frac{y}{4} mai i ngā taha e rua o te whārite.

$$\frac{1}{3}x=-\frac{1}{4}y-\frac{7}{12}$$

Me whakarea ngā taha e rua ki te 3.

$$x=3\left(-\frac{1}{4}y-\frac{7}{12}\right)$$

Whakareatia 3 ki te -\frac{y}{4}-\frac{7}{12}.

$$x=-\frac{3}{4}y-\frac{7}{4}$$

Whakakapia te \frac{-3y-7}{4} mō te x ki tērā atu whārite, \frac{1}{2}x+\frac{1}{3}y=-\frac{1}{6}.

$$\frac{1}{2}\left(-\frac{3}{4}y-\frac{7}{4}\right)+\frac{1}{3}y=-\frac{1}{6}$$

Whakareatia \frac{1}{2} ki te \frac{-3y-7}{4}.

$$-\frac{3}{8}y-\frac{7}{8}+\frac{1}{3}y=-\frac{1}{6}$$

Tāpiri -\frac{3y}{8} ki te \frac{y}{3}.

$$-\frac{1}{24}y-\frac{7}{8}=-\frac{1}{6}$$

Me tāpiri \frac{7}{8} ki ngā taha e rua o te whārite.

$$-\frac{1}{24}y=\frac{17}{24}$$

Me whakarea ngā taha e rua ki te -24.

$$y=-17$$

Whakaurua te -17 mō y ki x=-\frac{3}{4}y-\frac{7}{4}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{3}{4}\left(-17\right)-\frac{7}{4}$$

Whakareatia -\frac{3}{4} ki te -17.

$$x=\frac{51-7}{4}$$

Tāpiri -\frac{7}{4} ki te \frac{51}{4} 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.

$$x=11$$

Kua oti te pūnaha te whakatau.

$$x=11,y=-17$$

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.

$$\frac{1}{3}x+\frac{1}{4}y=-\frac{7}{12},\frac{1}{2}x+\frac{1}{3}y=-\frac{1}{6}$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}\frac{1}{3}&\frac{1}{4}\\\frac{1}{2}&\frac{1}{3}\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{7}{12}\\-\frac{1}{6}\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}\frac{1}{3}&\frac{1}{4}\\\frac{1}{2}&\frac{1}{3}\end{matrix}\right).

$$inverse(\left(\begin{matrix}\frac{1}{3}&\frac{1}{4}\\\frac{1}{2}&\frac{1}{3}\end{matrix}\right))\left(\begin{matrix}\frac{1}{3}&\frac{1}{4}\\\frac{1}{2}&\frac{1}{3}\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}\frac{1}{3}&\frac{1}{4}\\\frac{1}{2}&\frac{1}{3}\end{matrix}\right))\left(\begin{matrix}-\frac{7}{12}\\-\frac{1}{6}\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}\frac{1}{3}&\frac{1}{4}\\\frac{1}{2}&\frac{1}{3}\end{matrix}\right))\left(\begin{matrix}-\frac{7}{12}\\-\frac{1}{6}\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}\frac{1}{3}&\frac{1}{4}\\\frac{1}{2}&\frac{1}{3}\end{matrix}\right))\left(\begin{matrix}-\frac{7}{12}\\-\frac{1}{6}\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{\frac{1}{3}}{\frac{1}{3}\times \frac{1}{3}-\frac{1}{4}\times \frac{1}{2}}&-\frac{\frac{1}{4}}{\frac{1}{3}\times \frac{1}{3}-\frac{1}{4}\times \frac{1}{2}}\\-\frac{\frac{1}{2}}{\frac{1}{3}\times \frac{1}{3}-\frac{1}{4}\times \frac{1}{2}}&\frac{\frac{1}{3}}{\frac{1}{3}\times \frac{1}{3}-\frac{1}{4}\times \frac{1}{2}}\end{matrix}\right)\left(\begin{matrix}-\frac{7}{12}\\-\frac{1}{6}\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-24&18\\36&-24\end{matrix}\right)\left(\begin{matrix}-\frac{7}{12}\\-\frac{1}{6}\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-24\left(-\frac{7}{12}\right)+18\left(-\frac{1}{6}\right)\\36\left(-\frac{7}{12}\right)-24\left(-\frac{1}{6}\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}11\\-17\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=11,y=-17$$

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.

$$\frac{1}{3}x+\frac{1}{4}y=-\frac{7}{12},\frac{1}{2}x+\frac{1}{3}y=-\frac{1}{6}$$

Kia ōrite ai a \frac{x}{3} me \frac{x}{2}, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te \frac{1}{2} me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te \frac{1}{3}.

$$\frac{1}{2}\times \frac{1}{3}x+\frac{1}{2}\times \frac{1}{4}y=\frac{1}{2}\left(-\frac{7}{12}\right),\frac{1}{3}\times \frac{1}{2}x+\frac{1}{3}\times \frac{1}{3}y=\frac{1}{3}\left(-\frac{1}{6}\right)$$

Whakarūnātia.

$$\frac{1}{6}x+\frac{1}{8}y=-\frac{7}{24},\frac{1}{6}x+\frac{1}{9}y=-\frac{1}{18}$$

Me tango \frac{1}{6}x+\frac{1}{9}y=-\frac{1}{18} mai i \frac{1}{6}x+\frac{1}{8}y=-\frac{7}{24} mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$\frac{1}{6}x-\frac{1}{6}x+\frac{1}{8}y-\frac{1}{9}y=-\frac{7}{24}+\frac{1}{18}$$

Tāpiri \frac{x}{6} ki te -\frac{x}{6}. Ka whakakore atu ngā kupu \frac{x}{6} me -\frac{x}{6}, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$\frac{1}{8}y-\frac{1}{9}y=-\frac{7}{24}+\frac{1}{18}$$

Tāpiri \frac{y}{8} ki te -\frac{y}{9}.

$$\frac{1}{72}y=-\frac{7}{24}+\frac{1}{18}$$

Tāpiri -\frac{7}{24} ki te \frac{1}{18} 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.

$$\frac{1}{72}y=-\frac{17}{72}$$

Me whakarea ngā taha e rua ki te 72.

$$y=-17$$

Whakaurua te -17 mō y ki \frac{1}{2}x+\frac{1}{3}y=-\frac{1}{6}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$\frac{1}{2}x+\frac{1}{3}\left(-17\right)=-\frac{1}{6}$$

Whakareatia \frac{1}{3} ki te -17.

$$\frac{1}{2}x-\frac{17}{3}=-\frac{1}{6}$$

Me tāpiri \frac{17}{3} ki ngā taha e rua o te whārite.

$$\frac{1}{2}x=\frac{11}{2}$$

Me whakarea ngā taha e rua ki te 2.

$$x=11$$

Kua oti te pūnaha te whakatau.

$$x=11,y=-17$$