Whakaoti i te {l}{41x+53y=135}{53x+41y=147}

$$\left. \begin{array} { l } { 41 x + 53 y = 135 } \\ { 53 x + 41 y = 147 } \end{array} \right.$$

Whakaoti mō x, y

$$x=2$$

$$y=1$$

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.

$$41x+53y=135,53x+41y=147$$

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

$$41x+53y=135$$

Me tango 53y mai i ngā taha e rua o te whārite.

$$41x=-53y+135$$

Whakawehea ngā taha e rua ki te 41.

$$x=\frac{1}{41}\left(-53y+135\right)$$

Whakareatia \frac{1}{41} ki te -53y+135.

$$x=-\frac{53}{41}y+\frac{135}{41}$$

Whakakapia te \frac{-53y+135}{41} mō te x ki tērā atu whārite, 53x+41y=147.

$$53\left(-\frac{53}{41}y+\frac{135}{41}\right)+41y=147$$

Whakareatia 53 ki te \frac{-53y+135}{41}.

$$-\frac{2809}{41}y+\frac{7155}{41}+41y=147$$

Tāpiri -\frac{2809y}{41} ki te 41y.

$$-\frac{1128}{41}y+\frac{7155}{41}=147$$

Me tango \frac{7155}{41} mai i ngā taha e rua o te whārite.

$$-\frac{1128}{41}y=-\frac{1128}{41}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{1128}{41}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=1$$

Whakaurua te 1 mō y ki x=-\frac{53}{41}y+\frac{135}{41}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{-53+135}{41}$$

Tāpiri \frac{135}{41} ki te -\frac{53}{41} 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.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=1$$

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.

$$41x+53y=135,53x+41y=147$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}41&53\\53&41\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}135\\147\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}41&53\\53&41\end{matrix}\right).

$$inverse(\left(\begin{matrix}41&53\\53&41\end{matrix}\right))\left(\begin{matrix}41&53\\53&41\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}41&53\\53&41\end{matrix}\right))\left(\begin{matrix}135\\147\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}41&53\\53&41\end{matrix}\right))\left(\begin{matrix}135\\147\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}41&53\\53&41\end{matrix}\right))\left(\begin{matrix}135\\147\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{41}{41\times 41-53\times 53}&-\frac{53}{41\times 41-53\times 53}\\-\frac{53}{41\times 41-53\times 53}&\frac{41}{41\times 41-53\times 53}\end{matrix}\right)\left(\begin{matrix}135\\147\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{41}{1128}&\frac{53}{1128}\\\frac{53}{1128}&-\frac{41}{1128}\end{matrix}\right)\left(\begin{matrix}135\\147\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{41}{1128}\times 135+\frac{53}{1128}\times 147\\\frac{53}{1128}\times 135-\frac{41}{1128}\times 147\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2\\1\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=2,y=1$$

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.

$$41x+53y=135,53x+41y=147$$

Kia ōrite ai a 41x me 53x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 53 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 41.

$$53\times 41x+53\times 53y=53\times 135,41\times 53x+41\times 41y=41\times 147$$

Whakarūnātia.

$$2173x+2809y=7155,2173x+1681y=6027$$

Me tango 2173x+1681y=6027 mai i 2173x+2809y=7155 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2173x-2173x+2809y-1681y=7155-6027$$

Tāpiri 2173x ki te -2173x. Ka whakakore atu ngā kupu 2173x me -2173x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$2809y-1681y=7155-6027$$

Tāpiri 2809y ki te -1681y.

$$1128y=7155-6027$$

Tāpiri 7155 ki te -6027.

$$1128y=1128$$

Whakawehea ngā taha e rua ki te 1128.

$$y=1$$

Whakaurua te 1 mō y ki 53x+41y=147. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$53x+41=147$$

Me tango 41 mai i ngā taha e rua o te whārite.

$$53x=106$$

Whakawehea ngā taha e rua ki te 53.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=1$$