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

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

Whakaoti mō x, y

$$x=2$$

$$y=-1$$

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.

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

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

$$3x-2y-2=6$$

Me tāpiri 2 ki ngā taha e rua o te whārite.

$$3x-2y=8$$

Me tāpiri 2y ki ngā taha e rua o te whārite.

$$3x=2y+8$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(2y+8\right)$$

Whakareatia \frac{1}{3} ki te 8+2y.

$$x=\frac{2}{3}y+\frac{8}{3}$$

Whakakapia te \frac{8+2y}{3} mō te x ki tērā atu whārite, 3x+2y=4.

$$3\left(\frac{2}{3}y+\frac{8}{3}\right)+2y=4$$

Whakareatia 3 ki te \frac{8+2y}{3}.

$$2y+8+2y=4$$

Tāpiri 2y ki te 2y.

$$4y+8=4$$

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

$$4y=-4$$

Whakawehea ngā taha e rua ki te 4.

$$y=-1$$

Whakaurua te -1 mō y ki x=\frac{2}{3}y+\frac{8}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{2}{3}\left(-1\right)+\frac{8}{3}$$

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

$$x=\frac{-2+8}{3}$$

Tāpiri \frac{8}{3} ki te -\frac{2}{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.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=-1$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&-2\\3&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}8\\4\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}3&-2\\3&2\end{matrix}\right))\left(\begin{matrix}3&-2\\3&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-2\\3&2\end{matrix}\right))\left(\begin{matrix}8\\4\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-2\\3&2\end{matrix}\right))\left(\begin{matrix}8\\4\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-2\\3&2\end{matrix}\right))\left(\begin{matrix}8\\4\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{3\times 2-\left(-2\times 3\right)}&-\frac{-2}{3\times 2-\left(-2\times 3\right)}\\-\frac{3}{3\times 2-\left(-2\times 3\right)}&\frac{3}{3\times 2-\left(-2\times 3\right)}\end{matrix}\right)\left(\begin{matrix}8\\4\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{6}&\frac{1}{6}\\-\frac{1}{4}&\frac{1}{4}\end{matrix}\right)\left(\begin{matrix}8\\4\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{6}\times 8+\frac{1}{6}\times 4\\-\frac{1}{4}\times 8+\frac{1}{4}\times 4\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2\\-1\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=2,y=-1$$

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.

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

Me tango 3x+2y=4 mai i 3x-2y-2=6 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3x-3x-2y-2y-2=6-4$$

Tāpiri 3x ki te -3x. Ka whakakore atu ngā kupu 3x me -3x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-2y-2y-2=6-4$$

Tāpiri -2y ki te -2y.

$$-4y-2=6-4$$

Tāpiri 6 ki te -4.

$$-4y-2=2$$

Me tāpiri 2 ki ngā taha e rua o te whārite.

$$-4y=4$$

Whakawehea ngā taha e rua ki te -4.

$$y=-1$$

Whakaurua te -1 mō y ki 3x+2y=4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x+2\left(-1\right)=4$$

Whakareatia 2 ki te -1.

$$3x-2=4$$

Me tāpiri 2 ki ngā taha e rua o te whārite.

$$3x=6$$

Whakawehea ngā taha e rua ki te 3.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=-1$$