8x+2y=46,7x+3y=47

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.

8x+2y=46

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

8x=-2y+46

Me tango 2y mai i ngā taha e rua o te whārite.

x=\frac{1}{8}\left(-2y+46\right)

Whakawehea ngā taha e rua ki te 8.

x=-\frac{1}{4}y+\frac{23}{4}

Whakareatia \frac{1}{8} ki te -2y+46.

7\left(-\frac{1}{4}y+\frac{23}{4}\right)+3y=47

Whakakapia te \frac{-y+23}{4} mō te x ki tērā atu whārite, 7x+3y=47.

-\frac{7}{4}y+\frac{161}{4}+3y=47

Whakareatia 7 ki te \frac{-y+23}{4}.

\frac{5}{4}y+\frac{161}{4}=47

Tāpiri -\frac{7y}{4} ki te 3y.

\frac{5}{4}y=\frac{27}{4}

Me tango \frac{161}{4} mai i ngā taha e rua o te whārite.

y=\frac{27}{5}

Whakawehea ngā taha e rua o te whārite ki te \frac{5}{4}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

x=-\frac{1}{4}\times \frac{27}{5}+\frac{23}{4}

Whakaurua te \frac{27}{5} mō y ki x=-\frac{1}{4}y+\frac{23}{4}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

x=-\frac{27}{20}+\frac{23}{4}

Whakareatia -\frac{1}{4} ki te \frac{27}{5} 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{22}{5}

Tāpiri \frac{23}{4} ki te -\frac{27}{20} 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{22}{5},y=\frac{27}{5}

Kua oti te pūnaha te whakatau.

8x+2y=46,7x+3y=47

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

\left(\begin{matrix}8&2\\7&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}46\\47\end{matrix}\right)

Tuhia ngā whārite ki te tikanga tātai poukapa.

inverse(\left(\begin{matrix}8&2\\7&3\end{matrix}\right))\left(\begin{matrix}8&2\\7&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&2\\7&3\end{matrix}\right))\left(\begin{matrix}46\\47\end{matrix}\right)

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}8&2\\7&3\end{matrix}\right).

\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&2\\7&3\end{matrix}\right))\left(\begin{matrix}46\\47\end{matrix}\right)

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

\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&2\\7&3\end{matrix}\right))\left(\begin{matrix}46\\47\end{matrix}\right)

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

\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{8\times 3-2\times 7}&-\frac{2}{8\times 3-2\times 7}\\-\frac{7}{8\times 3-2\times 7}&\frac{8}{8\times 3-2\times 7}\end{matrix}\right)\left(\begin{matrix}46\\47\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{3}{10}&-\frac{1}{5}\\-\frac{7}{10}&\frac{4}{5}\end{matrix}\right)\left(\begin{matrix}46\\47\end{matrix}\right)

Mahia ngā tātaitanga.

\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{10}\times 46-\frac{1}{5}\times 47\\-\frac{7}{10}\times 46+\frac{4}{5}\times 47\end{matrix}\right)

Whakareatia ngā poukapa.

\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{22}{5}\\\frac{27}{5}\end{matrix}\right)

Mahia ngā tātaitanga.

x=\frac{22}{5},y=\frac{27}{5}

Tangohia ngā huānga poukapa x me y.

8x+2y=46,7x+3y=47

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.

7\times 8x+7\times 2y=7\times 46,8\times 7x+8\times 3y=8\times 47

Kia ōrite ai a 8x me 7x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 7 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 8.

56x+14y=322,56x+24y=376

Whakarūnātia.

56x-56x+14y-24y=322-376

Me tango 56x+24y=376 mai i 56x+14y=322 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

14y-24y=322-376

Tāpiri 56x ki te -56x. Ka whakakore atu ngā kupu 56x me -56x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

-10y=322-376

Tāpiri 14y ki te -24y.

-10y=-54

Tāpiri 322 ki te -376.

y=\frac{27}{5}

Whakawehea ngā taha e rua ki te -10.

7x+3\times \frac{27}{5}=47

Whakaurua te \frac{27}{5} mō y ki 7x+3y=47. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

7x+\frac{81}{5}=47

Whakareatia 3 ki te \frac{27}{5}.

7x=\frac{154}{5}

Me tango \frac{81}{5} mai i ngā taha e rua o te whārite.

x=\frac{22}{5}

Whakawehea ngā taha e rua ki te 7.

x=\frac{22}{5},y=\frac{27}{5}

Kua oti te pūnaha te whakatau.