Whakaoti i te {l}{3x-2y=6}{x/3-y/6=1/2}

$$\left. \begin{array} { l } { 3 x - 2 y = 6 } \\ { \frac { x } { 3 } - \frac { y } { 6 } = \frac { 1 } { 2 } } \end{array} \right.$$

Whakaoti mō x, y

$$x=0$$

$$y=-3$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Me whakarea ngā taha e rua o te whārite ki te 6, arā, te tauraro pātahi he tino iti rawa te kitea o 3,6,2.

$$2x-y=3$$

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.

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

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

$$3x-2y=6$$

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

$$3x=2y+6$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(2y+6\right)$$

Whakareatia \frac{1}{3} ki te 6+2y.

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

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

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

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

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

Tāpiri \frac{4y}{3} ki te -y.

$$\frac{1}{3}y+4=3$$

Me tango 4 mai i ngā taha e rua o te whārite.

$$\frac{1}{3}y=-1$$

Me whakarea ngā taha e rua ki te 3.

$$y=-3$$

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

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

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

$$x=-2+2$$

Tāpiri 2 ki te -2.

$$x=0$$

Kua oti te pūnaha te whakatau.

$$x=0,y=-3$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Me whakarea ngā taha e rua o te whārite ki te 6, arā, te tauraro pātahi he tino iti rawa te kitea o 3,6,2.

$$2x-y=3$$

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

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&-2\\2&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}6\\3\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&-2\\2&-1\end{matrix}\right).

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

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

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

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

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

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{3\left(-1\right)-\left(-2\times 2\right)}&-\frac{-2}{3\left(-1\right)-\left(-2\times 2\right)}\\-\frac{2}{3\left(-1\right)-\left(-2\times 2\right)}&\frac{3}{3\left(-1\right)-\left(-2\times 2\right)}\end{matrix}\right)\left(\begin{matrix}6\\3\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-1&2\\-2&3\end{matrix}\right)\left(\begin{matrix}6\\3\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-6+2\times 3\\-2\times 6+3\times 3\end{matrix}\right)$$

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

Whakaarohia te whārite tuarua. Me whakarea ngā taha e rua o te whārite ki te 6, arā, te tauraro pātahi he tino iti rawa te kitea o 3,6,2.

$$2x-y=3$$

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.

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

Kia ōrite ai a 3x me 2x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$2\times 3x+2\left(-2\right)y=2\times 6,3\times 2x+3\left(-1\right)y=3\times 3$$

Whakarūnātia.

$$6x-4y=12,6x-3y=9$$

Me tango 6x-3y=9 mai i 6x-4y=12 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x-4y+3y=12-9$$

Tāpiri 6x ki te -6x. Ka whakakore atu ngā kupu 6x me -6x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-4y+3y=12-9$$

Tāpiri -4y ki te 3y.

$$-y=12-9$$

Tāpiri 12 ki te -9.

$$-y=3$$

Whakawehea ngā taha e rua ki te -1.

$$y=-3$$

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

$$2x-\left(-3\right)=3$$

Me tango 3 mai i ngā taha e rua o te whārite.

$$2x=0$$

Whakawehea ngā taha e rua ki te 2.

$$x=0$$

Kua oti te pūnaha te whakatau.

$$x=0,y=-3$$