Whakaoti i te {8x+2y=46}{7x+3y=47}

$$\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.$$

Whakaoti mō x, y

$$x = \frac{22}{5} = 4\frac{2}{5} = 4.4$$

$$y = \frac{27}{5} = 5\frac{2}{5} = 5.4$$

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.

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

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.

$$8x=-2y+46$$

Whakawehea ngā taha e rua ki te 8.

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

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

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

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

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

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

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

Tāpiri -\frac{7y}{4} ki te 3y.

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

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

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

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.

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

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.

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

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.

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

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.

$$x=\frac{22}{5}$$

Kua oti te pūnaha te whakatau.

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

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

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

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.

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

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.

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

Whakarūnātia.

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

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.

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

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.

$$14y-24y=322-376$$

Tāpiri 14y ki te -24y.

$$-10y=322-376$$

Tāpiri 322 ki te -376.

$$-10y=-54$$

Whakawehea ngā taha e rua ki te -10.

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

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.

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

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

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

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

$$7x=\frac{154}{5}$$

Whakawehea ngā taha e rua ki te 7.

$$x=\frac{22}{5}$$

Kua oti te pūnaha te whakatau.

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