Whakaoti i te {l}{x/92+y=58}{x-y/79=57}

$$\left. \begin{array} { l } { \frac { x } { 92 } + y = 58 } \\ { x - \frac { y } { 79 } = 57 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{419612}{7269} = 57\frac{5279}{7269} \approx 57.726234695$$

$$y = \frac{417041}{7269} = 57\frac{2708}{7269} \approx 57.372540927$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Whakareatia ngā taha e rua o te whārite ki te 92.

$$x+92y=5336$$

Whakaarohia te whārite tuarua. Whakareatia ngā taha e rua o te whārite ki te 79.

$$79x-y=4503$$

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.

$$x+92y=5336,79x-y=4503$$

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

$$x+92y=5336$$

Me tango 92y mai i ngā taha e rua o te whārite.

$$x=-92y+5336$$

Whakakapia te -92y+5336 mō te x ki tērā atu whārite, 79x-y=4503.

$$79\left(-92y+5336\right)-y=4503$$

Whakareatia 79 ki te -92y+5336.

$$-7268y+421544-y=4503$$

Tāpiri -7268y ki te -y.

$$-7269y+421544=4503$$

Me tango 421544 mai i ngā taha e rua o te whārite.

$$-7269y=-417041$$

Whakawehea ngā taha e rua ki te -7269.

$$y=\frac{417041}{7269}$$

Whakaurua te \frac{417041}{7269} mō y ki x=-92y+5336. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-92\times \frac{417041}{7269}+5336$$

Whakareatia -92 ki te \frac{417041}{7269}.

$$x=-\frac{38367772}{7269}+5336$$

Tāpiri 5336 ki te -\frac{38367772}{7269}.

$$x=\frac{419612}{7269}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{419612}{7269},y=\frac{417041}{7269}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Whakareatia ngā taha e rua o te whārite ki te 92.

$$x+92y=5336$$

Whakaarohia te whārite tuarua. Whakareatia ngā taha e rua o te whārite ki te 79.

$$79x-y=4503$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$x+92y=5336,79x-y=4503$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&92\\79&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5336\\4503\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&92\\79&-1\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&92\\79&-1\end{matrix}\right))\left(\begin{matrix}1&92\\79&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&92\\79&-1\end{matrix}\right))\left(\begin{matrix}5336\\4503\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&92\\79&-1\end{matrix}\right))\left(\begin{matrix}5336\\4503\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&92\\79&-1\end{matrix}\right))\left(\begin{matrix}5336\\4503\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{-1-92\times 79}&-\frac{92}{-1-92\times 79}\\-\frac{79}{-1-92\times 79}&\frac{1}{-1-92\times 79}\end{matrix}\right)\left(\begin{matrix}5336\\4503\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{7269}&\frac{92}{7269}\\\frac{79}{7269}&-\frac{1}{7269}\end{matrix}\right)\left(\begin{matrix}5336\\4503\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{7269}\times 5336+\frac{92}{7269}\times 4503\\\frac{79}{7269}\times 5336-\frac{1}{7269}\times 4503\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{419612}{7269}\\\frac{417041}{7269}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{419612}{7269},y=\frac{417041}{7269}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Whakareatia ngā taha e rua o te whārite ki te 92.

$$x+92y=5336$$

Whakaarohia te whārite tuarua. Whakareatia ngā taha e rua o te whārite ki te 79.

$$79x-y=4503$$

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.

$$x+92y=5336,79x-y=4503$$

Kia ōrite ai a x me 79x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 79 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$79x+79\times 92y=79\times 5336,79x-y=4503$$

Whakarūnātia.

$$79x+7268y=421544,79x-y=4503$$

Me tango 79x-y=4503 mai i 79x+7268y=421544 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$79x-79x+7268y+y=421544-4503$$

Tāpiri 79x ki te -79x. Ka whakakore atu ngā kupu 79x me -79x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$7268y+y=421544-4503$$

Tāpiri 7268y ki te y.

$$7269y=421544-4503$$

Tāpiri 421544 ki te -4503.

$$7269y=417041$$

Whakawehea ngā taha e rua ki te 7269.

$$y=\frac{417041}{7269}$$

Whakaurua te \frac{417041}{7269} mō y ki 79x-y=4503. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$79x-\frac{417041}{7269}=4503$$

Me tāpiri \frac{417041}{7269} ki ngā taha e rua o te whārite.

$$79x=\frac{33149348}{7269}$$

Whakawehea ngā taha e rua ki te 79.

$$x=\frac{419612}{7269}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{419612}{7269},y=\frac{417041}{7269}$$