Whakaoti i te 3^14x+2y=6,3x+6y=18

$$3 ^ { 14 } x + 2 y = 6,3 x + 6 y = 18$$

Whakaoti mō x, y

$$x=0$$

$$y=3$$

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.

$$4782969x+2y=6,3x+6y=18$$

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

$$4782969x+2y=6$$

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

$$4782969x=-2y+6$$

Whakawehea ngā taha e rua ki te 4782969.

$$x=\frac{1}{4782969}\left(-2y+6\right)$$

Whakareatia \frac{1}{4782969} ki te -2y+6.

$$x=-\frac{2}{4782969}y+\frac{2}{1594323}$$

Whakakapia te -\frac{2y}{4782969}+\frac{2}{1594323} mō te x ki tērā atu whārite, 3x+6y=18.

$$3\left(-\frac{2}{4782969}y+\frac{2}{1594323}\right)+6y=18$$

Whakareatia 3 ki te -\frac{2y}{4782969}+\frac{2}{1594323}.

$$-\frac{2}{1594323}y+\frac{2}{531441}+6y=18$$

Tāpiri -\frac{2y}{1594323} ki te 6y.

$$\frac{9565936}{1594323}y+\frac{2}{531441}=18$$

Me tango \frac{2}{531441} mai i ngā taha e rua o te whārite.

$$\frac{9565936}{1594323}y=\frac{9565936}{531441}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{9565936}{1594323}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=3$$

Whakaurua te 3 mō y ki x=-\frac{2}{4782969}y+\frac{2}{1594323}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{2}{4782969}\times 3+\frac{2}{1594323}$$

Whakareatia -\frac{2}{4782969} ki te 3.

$$x=\frac{-2+2}{1594323}$$

Tāpiri \frac{2}{1594323} ki te -\frac{2}{1594323} 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.

$$x=0$$

Kua oti te pūnaha te whakatau.

$$x=0,y=3$$

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.

$$4782969x+2y=6,3x+6y=18$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4782969&2\\3&6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}6\\18\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4782969&2\\3&6\end{matrix}\right).

$$inverse(\left(\begin{matrix}4782969&2\\3&6\end{matrix}\right))\left(\begin{matrix}4782969&2\\3&6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4782969&2\\3&6\end{matrix}\right))\left(\begin{matrix}6\\18\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4782969&2\\3&6\end{matrix}\right))\left(\begin{matrix}6\\18\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4782969&2\\3&6\end{matrix}\right))\left(\begin{matrix}6\\18\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{6}{4782969\times 6-2\times 3}&-\frac{2}{4782969\times 6-2\times 3}\\-\frac{3}{4782969\times 6-2\times 3}&\frac{4782969}{4782969\times 6-2\times 3}\end{matrix}\right)\left(\begin{matrix}6\\18\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4782968}&-\frac{1}{14348904}\\-\frac{1}{9565936}&\frac{1594323}{9565936}\end{matrix}\right)\left(\begin{matrix}6\\18\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4782968}\times 6-\frac{1}{14348904}\times 18\\-\frac{1}{9565936}\times 6+\frac{1594323}{9565936}\times 18\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}0\\3\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=0,y=3$$

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.

$$4782969x+2y=6,3x+6y=18$$

Kia ōrite ai a 4782969x me 3x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 3 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 4782969.

$$3\times 4782969x+3\times 2y=3\times 6,4782969\times 3x+4782969\times 6y=4782969\times 18$$

Whakarūnātia.

$$14348907x+6y=18,14348907x+28697814y=86093442$$

Me tango 14348907x+28697814y=86093442 mai i 14348907x+6y=18 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$14348907x-14348907x+6y-28697814y=18-86093442$$

Tāpiri 14348907x ki te -14348907x. Ka whakakore atu ngā kupu 14348907x me -14348907x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$6y-28697814y=18-86093442$$

Tāpiri 6y ki te -28697814y.

$$-28697808y=18-86093442$$

Tāpiri 18 ki te -86093442.

$$-28697808y=-86093424$$

Whakawehea ngā taha e rua ki te -28697808.

$$y=3$$

Whakaurua te 3 mō y ki 3x+6y=18. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x+6\times 3=18$$

Whakareatia 6 ki te 3.

$$3x+18=18$$

Me tango 18 mai i ngā taha e rua o te whārite.

$$3x=0$$

Whakawehea ngā taha e rua ki te 3.

$$x=0$$

Kua oti te pūnaha te whakatau.

$$x=0,y=3$$