Whakaoti i te {l}{2x+5y=259}{199x-2y=1127}

$$\left. \begin{array} { l } { 2 x + 5 y = 259 } \\ { 199 x - 2 y = 1127 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{2051}{333} = 6\frac{53}{333} \approx 6.159159159$$

$$y = \frac{16429}{333} = 49\frac{112}{333} \approx 49.336336336$$

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.

$$2x+5y=259,199x-2y=1127$$

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

$$2x+5y=259$$

Me tango 5y mai i ngā taha e rua o te whārite.

$$2x=-5y+259$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(-5y+259\right)$$

Whakareatia \frac{1}{2} ki te -5y+259.

$$x=-\frac{5}{2}y+\frac{259}{2}$$

Whakakapia te \frac{-5y+259}{2} mō te x ki tērā atu whārite, 199x-2y=1127.

$$199\left(-\frac{5}{2}y+\frac{259}{2}\right)-2y=1127$$

Whakareatia 199 ki te \frac{-5y+259}{2}.

$$-\frac{995}{2}y+\frac{51541}{2}-2y=1127$$

Tāpiri -\frac{995y}{2} ki te -2y.

$$-\frac{999}{2}y+\frac{51541}{2}=1127$$

Me tango \frac{51541}{2} mai i ngā taha e rua o te whārite.

$$-\frac{999}{2}y=-\frac{49287}{2}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{999}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{16429}{333}$$

Whakaurua te \frac{16429}{333} mō y ki x=-\frac{5}{2}y+\frac{259}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{5}{2}\times \frac{16429}{333}+\frac{259}{2}$$

Whakareatia -\frac{5}{2} ki te \frac{16429}{333} 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.

$$x=-\frac{82145}{666}+\frac{259}{2}$$

Tāpiri \frac{259}{2} ki te -\frac{82145}{666} 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.

$$x=\frac{2051}{333}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{2051}{333},y=\frac{16429}{333}$$

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.

$$2x+5y=259,199x-2y=1127$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&5\\199&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}259\\1127\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&5\\199&-2\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&5\\199&-2\end{matrix}\right))\left(\begin{matrix}2&5\\199&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&5\\199&-2\end{matrix}\right))\left(\begin{matrix}259\\1127\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&5\\199&-2\end{matrix}\right))\left(\begin{matrix}259\\1127\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&5\\199&-2\end{matrix}\right))\left(\begin{matrix}259\\1127\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{2\left(-2\right)-5\times 199}&-\frac{5}{2\left(-2\right)-5\times 199}\\-\frac{199}{2\left(-2\right)-5\times 199}&\frac{2}{2\left(-2\right)-5\times 199}\end{matrix}\right)\left(\begin{matrix}259\\1127\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{999}&\frac{5}{999}\\\frac{199}{999}&-\frac{2}{999}\end{matrix}\right)\left(\begin{matrix}259\\1127\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{999}\times 259+\frac{5}{999}\times 1127\\\frac{199}{999}\times 259-\frac{2}{999}\times 1127\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2051}{333}\\\frac{16429}{333}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{2051}{333},y=\frac{16429}{333}$$

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.

$$2x+5y=259,199x-2y=1127$$

Kia ōrite ai a 2x me 199x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 199 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

$$199\times 2x+199\times 5y=199\times 259,2\times 199x+2\left(-2\right)y=2\times 1127$$

Whakarūnātia.

$$398x+995y=51541,398x-4y=2254$$

Me tango 398x-4y=2254 mai i 398x+995y=51541 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$398x-398x+995y+4y=51541-2254$$

Tāpiri 398x ki te -398x. Ka whakakore atu ngā kupu 398x me -398x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$995y+4y=51541-2254$$

Tāpiri 995y ki te 4y.

$$999y=51541-2254$$

Tāpiri 51541 ki te -2254.

$$999y=49287$$

Whakawehea ngā taha e rua ki te 999.

$$y=\frac{16429}{333}$$

Whakaurua te \frac{16429}{333} mō y ki 199x-2y=1127. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$199x-2\times \frac{16429}{333}=1127$$

Whakareatia -2 ki te \frac{16429}{333}.

$$199x-\frac{32858}{333}=1127$$

Me tāpiri \frac{32858}{333} ki ngā taha e rua o te whārite.

$$199x=\frac{408149}{333}$$

Whakawehea ngā taha e rua ki te 199.

$$x=\frac{2051}{333}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{2051}{333},y=\frac{16429}{333}$$