Whakaoti i te {r}{x+1/2+y-1/3=8}{x-1/3+y+1/2=9}

$$\left. \begin{array} { r } { \frac { x + 1 } { 2 } + \frac { y - 1 } { 3 } = 8 } \\ { \frac { x - 1 } { 3 } + \frac { y + 1 } { 2 } = 9 } \end{array} \right.$$

Whakaoti mō x, y

$$x=7$$

$$y=13$$

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.

$$\frac{1}{2}\left(x+1\right)+\frac{1}{3}\left(y-1\right)=8,\frac{1}{3}\left(x-1\right)+\frac{1}{2}\left(y+1\right)=9$$

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

$$\frac{1}{2}\left(x+1\right)+\frac{1}{3}\left(y-1\right)=8$$

Whakareatia \frac{1}{2} ki te x+1.

$$\frac{1}{2}x+\frac{1}{2}+\frac{1}{3}\left(y-1\right)=8$$

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

$$\frac{1}{2}x+\frac{1}{2}+\frac{1}{3}y-\frac{1}{3}=8$$

Tāpiri \frac{1}{2} ki te -\frac{1}{3} 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.

$$\frac{1}{2}x+\frac{1}{3}y+\frac{1}{6}=8$$

Me tango \frac{1}{6} mai i ngā taha e rua o te whārite.

$$\frac{1}{2}x+\frac{1}{3}y=\frac{47}{6}$$

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

$$\frac{1}{2}x=-\frac{1}{3}y+\frac{47}{6}$$

Me whakarea ngā taha e rua ki te 2.

$$x=2\left(-\frac{1}{3}y+\frac{47}{6}\right)$$

Whakareatia 2 ki te -\frac{y}{3}+\frac{47}{6}.

$$x=-\frac{2}{3}y+\frac{47}{3}$$

Whakakapia te \frac{-2y+47}{3} mō te x ki tērā atu whārite, \frac{1}{3}\left(x-1\right)+\frac{1}{2}\left(y+1\right)=9.

$$\frac{1}{3}\left(-\frac{2}{3}y+\frac{47}{3}-1\right)+\frac{1}{2}\left(y+1\right)=9$$

Tāpiri \frac{47}{3} ki te -1.

$$\frac{1}{3}\left(-\frac{2}{3}y+\frac{44}{3}\right)+\frac{1}{2}\left(y+1\right)=9$$

Whakareatia \frac{1}{3} ki te \frac{-2y+44}{3}.

$$-\frac{2}{9}y+\frac{44}{9}+\frac{1}{2}\left(y+1\right)=9$$

Whakareatia \frac{1}{2} ki te y+1.

$$-\frac{2}{9}y+\frac{44}{9}+\frac{1}{2}y+\frac{1}{2}=9$$

Tāpiri -\frac{2y}{9} ki te \frac{y}{2}.

$$\frac{5}{18}y+\frac{44}{9}+\frac{1}{2}=9$$

Tāpiri \frac{44}{9} ki te \frac{1}{2} 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.

$$\frac{5}{18}y+\frac{97}{18}=9$$

Me tango \frac{97}{18} mai i ngā taha e rua o te whārite.

$$\frac{5}{18}y=\frac{65}{18}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{5}{18}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=13$$

Whakaurua te 13 mō y ki x=-\frac{2}{3}y+\frac{47}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{2}{3}\times 13+\frac{47}{3}$$

Whakareatia -\frac{2}{3} ki te 13.

$$x=\frac{-26+47}{3}$$

Tāpiri \frac{47}{3} ki te -\frac{26}{3} 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.

$$x=7$$

Kua oti te pūnaha te whakatau.

$$x=7,y=13$$

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.

$$\frac{1}{2}\left(x+1\right)+\frac{1}{3}\left(y-1\right)=8,\frac{1}{3}\left(x-1\right)+\frac{1}{2}\left(y+1\right)=9$$

Whakarūnātia te whārite tuatahi ki te āhua tānga ngahuru.

$$\frac{1}{2}\left(x+1\right)+\frac{1}{3}\left(y-1\right)=8$$

Whakareatia \frac{1}{2} ki te x+1.

$$\frac{1}{2}x+\frac{1}{2}+\frac{1}{3}\left(y-1\right)=8$$

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

$$\frac{1}{2}x+\frac{1}{2}+\frac{1}{3}y-\frac{1}{3}=8$$

Tāpiri \frac{1}{2} ki te -\frac{1}{3} 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.

$$\frac{1}{2}x+\frac{1}{3}y+\frac{1}{6}=8$$

Me tango \frac{1}{6} mai i ngā taha e rua o te whārite.

$$\frac{1}{2}x+\frac{1}{3}y=\frac{47}{6}$$

Whakarūnātia te whārite tuarua ki te āhua tānga ngahuru.

$$\frac{1}{3}\left(x-1\right)+\frac{1}{2}\left(y+1\right)=9$$

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

$$\frac{1}{3}x-\frac{1}{3}+\frac{1}{2}\left(y+1\right)=9$$

Whakareatia \frac{1}{2} ki te y+1.

$$\frac{1}{3}x-\frac{1}{3}+\frac{1}{2}y+\frac{1}{2}=9$$

Tāpiri -\frac{1}{3} ki te \frac{1}{2} 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.

$$\frac{1}{3}x+\frac{1}{2}y+\frac{1}{6}=9$$

Me tango \frac{1}{6} mai i ngā taha e rua o te whārite.

$$\frac{1}{3}x+\frac{1}{2}y=\frac{53}{6}$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}\frac{1}{2}&\frac{1}{3}\\\frac{1}{3}&\frac{1}{2}\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{47}{6}\\\frac{53}{6}\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}\frac{1}{2}&\frac{1}{3}\\\frac{1}{3}&\frac{1}{2}\end{matrix}\right))\left(\begin{matrix}\frac{1}{2}&\frac{1}{3}\\\frac{1}{3}&\frac{1}{2}\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}\frac{1}{2}&\frac{1}{3}\\\frac{1}{3}&\frac{1}{2}\end{matrix}\right))\left(\begin{matrix}\frac{47}{6}\\\frac{53}{6}\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}\frac{1}{2}&\frac{1}{3}\\\frac{1}{3}&\frac{1}{2}\end{matrix}\right))\left(\begin{matrix}\frac{47}{6}\\\frac{53}{6}\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}\frac{1}{2}&\frac{1}{3}\\\frac{1}{3}&\frac{1}{2}\end{matrix}\right))\left(\begin{matrix}\frac{47}{6}\\\frac{53}{6}\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{\frac{1}{2}}{\frac{1}{2}\times \frac{1}{2}-\frac{1}{3}\times \frac{1}{3}}&-\frac{\frac{1}{3}}{\frac{1}{2}\times \frac{1}{2}-\frac{1}{3}\times \frac{1}{3}}\\-\frac{\frac{1}{3}}{\frac{1}{2}\times \frac{1}{2}-\frac{1}{3}\times \frac{1}{3}}&\frac{\frac{1}{2}}{\frac{1}{2}\times \frac{1}{2}-\frac{1}{3}\times \frac{1}{3}}\end{matrix}\right)\left(\begin{matrix}\frac{47}{6}\\\frac{53}{6}\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{18}{5}&-\frac{12}{5}\\-\frac{12}{5}&\frac{18}{5}\end{matrix}\right)\left(\begin{matrix}\frac{47}{6}\\\frac{53}{6}\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{18}{5}\times \frac{47}{6}-\frac{12}{5}\times \frac{53}{6}\\-\frac{12}{5}\times \frac{47}{6}+\frac{18}{5}\times \frac{53}{6}\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}7\\13\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=7,y=13$$