Whakaoti i te {l}{3x+2y=4}{3y+6x=10}

$$\left. \begin{array} { l } { 3 x + 2 y = 4 } \\ { 3 y + 6 x = 10 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{8}{3} = 2\frac{2}{3} \approx 2.666666667$$

$$y=-2$$

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.

$$3x+2y=4,6x+3y=10$$

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

$$3x+2y=4$$

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

$$3x=-2y+4$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(-2y+4\right)$$

Whakareatia \frac{1}{3} ki te -2y+4.

$$x=-\frac{2}{3}y+\frac{4}{3}$$

Whakakapia te \frac{-2y+4}{3} mō te x ki tērā atu whārite, 6x+3y=10.

$$6\left(-\frac{2}{3}y+\frac{4}{3}\right)+3y=10$$

Whakareatia 6 ki te \frac{-2y+4}{3}.

$$-4y+8+3y=10$$

Tāpiri -4y ki te 3y.

$$-y+8=10$$

Me tango 8 mai i ngā taha e rua o te whārite.

$$-y=2$$

Whakawehea ngā taha e rua ki te -1.

$$y=-2$$

Whakaurua te -2 mō y ki x=-\frac{2}{3}y+\frac{4}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{2}{3}\left(-2\right)+\frac{4}{3}$$

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

$$x=\frac{4+4}{3}$$

Tāpiri \frac{4}{3} ki te \frac{4}{3} 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.

$$x=\frac{8}{3}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{8}{3},y=-2$$

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.

$$3x+2y=4,6x+3y=10$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&2\\6&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\\10\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}3&2\\6&3\end{matrix}\right))\left(\begin{matrix}3&2\\6&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&2\\6&3\end{matrix}\right))\left(\begin{matrix}4\\10\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&2\\6&3\end{matrix}\right))\left(\begin{matrix}4\\10\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&2\\6&3\end{matrix}\right))\left(\begin{matrix}4\\10\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{3\times 3-2\times 6}&-\frac{2}{3\times 3-2\times 6}\\-\frac{6}{3\times 3-2\times 6}&\frac{3}{3\times 3-2\times 6}\end{matrix}\right)\left(\begin{matrix}4\\10\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-1&\frac{2}{3}\\2&-1\end{matrix}\right)\left(\begin{matrix}4\\10\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-4+\frac{2}{3}\times 10\\2\times 4-10\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{8}{3}\\-2\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{8}{3},y=-2$$

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.

$$3x+2y=4,6x+3y=10$$

Kia ōrite ai a 3x me 6x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 6 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$6\times 3x+6\times 2y=6\times 4,3\times 6x+3\times 3y=3\times 10$$

Whakarūnātia.

$$18x+12y=24,18x+9y=30$$

Me tango 18x+9y=30 mai i 18x+12y=24 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$18x-18x+12y-9y=24-30$$

Tāpiri 18x ki te -18x. Ka whakakore atu ngā kupu 18x me -18x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$12y-9y=24-30$$

Tāpiri 12y ki te -9y.

$$3y=24-30$$

Tāpiri 24 ki te -30.

$$3y=-6$$

Whakawehea ngā taha e rua ki te 3.

$$y=-2$$

Whakaurua te -2 mō y ki 6x+3y=10. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$6x+3\left(-2\right)=10$$

Whakareatia 3 ki te -2.

$$6x-6=10$$

Me tāpiri 6 ki ngā taha e rua o te whārite.

$$6x=16$$

Whakawehea ngā taha e rua ki te 6.

$$x=\frac{8}{3}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{8}{3},y=-2$$