Whakaoti i te {l}{23x+7y=13}{101x+73y=1}

$$\left. \begin{array} { l } { 23 x + 7 y = 13 } \\ { 101 x + 73 y = 1 } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{157}{162}\approx 0.969135802$$

$$y = -\frac{215}{162} = -1\frac{53}{162} \approx -1.327160494$$

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.

$$23x+7y=13,101x+73y=1$$

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

$$23x+7y=13$$

Me tango 7y mai i ngā taha e rua o te whārite.

$$23x=-7y+13$$

Whakawehea ngā taha e rua ki te 23.

$$x=\frac{1}{23}\left(-7y+13\right)$$

Whakareatia \frac{1}{23} ki te -7y+13.

$$x=-\frac{7}{23}y+\frac{13}{23}$$

Whakakapia te \frac{-7y+13}{23} mō te x ki tērā atu whārite, 101x+73y=1.

$$101\left(-\frac{7}{23}y+\frac{13}{23}\right)+73y=1$$

Whakareatia 101 ki te \frac{-7y+13}{23}.

$$-\frac{707}{23}y+\frac{1313}{23}+73y=1$$

Tāpiri -\frac{707y}{23} ki te 73y.

$$\frac{972}{23}y+\frac{1313}{23}=1$$

Me tango \frac{1313}{23} mai i ngā taha e rua o te whārite.

$$\frac{972}{23}y=-\frac{1290}{23}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{972}{23}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-\frac{215}{162}$$

Whakaurua te -\frac{215}{162} mō y ki x=-\frac{7}{23}y+\frac{13}{23}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{7}{23}\left(-\frac{215}{162}\right)+\frac{13}{23}$$

Whakareatia -\frac{7}{23} ki te -\frac{215}{162} 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.

$$x=\frac{1505}{3726}+\frac{13}{23}$$

Tāpiri \frac{13}{23} ki te \frac{1505}{3726} 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.

$$x=\frac{157}{162}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{157}{162},y=-\frac{215}{162}$$

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.

$$23x+7y=13,101x+73y=1$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}23&7\\101&73\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}13\\1\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}23&7\\101&73\end{matrix}\right).

$$inverse(\left(\begin{matrix}23&7\\101&73\end{matrix}\right))\left(\begin{matrix}23&7\\101&73\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}23&7\\101&73\end{matrix}\right))\left(\begin{matrix}13\\1\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}23&7\\101&73\end{matrix}\right))\left(\begin{matrix}13\\1\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}23&7\\101&73\end{matrix}\right))\left(\begin{matrix}13\\1\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{73}{23\times 73-7\times 101}&-\frac{7}{23\times 73-7\times 101}\\-\frac{101}{23\times 73-7\times 101}&\frac{23}{23\times 73-7\times 101}\end{matrix}\right)\left(\begin{matrix}13\\1\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{73}{972}&-\frac{7}{972}\\-\frac{101}{972}&\frac{23}{972}\end{matrix}\right)\left(\begin{matrix}13\\1\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{73}{972}\times 13-\frac{7}{972}\\-\frac{101}{972}\times 13+\frac{23}{972}\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{157}{162}\\-\frac{215}{162}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{157}{162},y=-\frac{215}{162}$$

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.

$$23x+7y=13,101x+73y=1$$

Kia ōrite ai a 23x me 101x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 101 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 23.

$$101\times 23x+101\times 7y=101\times 13,23\times 101x+23\times 73y=23$$

Whakarūnātia.

$$2323x+707y=1313,2323x+1679y=23$$

Me tango 2323x+1679y=23 mai i 2323x+707y=1313 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2323x-2323x+707y-1679y=1313-23$$

Tāpiri 2323x ki te -2323x. Ka whakakore atu ngā kupu 2323x me -2323x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$707y-1679y=1313-23$$

Tāpiri 707y ki te -1679y.

$$-972y=1313-23$$

Tāpiri 1313 ki te -23.

$$-972y=1290$$

Whakawehea ngā taha e rua ki te -972.

$$y=-\frac{215}{162}$$

Whakaurua te -\frac{215}{162} mō y ki 101x+73y=1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$101x+73\left(-\frac{215}{162}\right)=1$$

Whakareatia 73 ki te -\frac{215}{162}.

$$101x-\frac{15695}{162}=1$$

Me tāpiri \frac{15695}{162} ki ngā taha e rua o te whārite.

$$101x=\frac{15857}{162}$$

Whakawehea ngā taha e rua ki te 101.

$$x=\frac{157}{162}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{157}{162},y=-\frac{215}{162}$$