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

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

Whakaoti mō x, y

$$x=0$$

$$y=-2$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Whakareatia ngā taha e rua o te whārite ki te 3.

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

Pahekotia te y me -3y, ka -2y.

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

Tangohia te 2 mai i ngā taha e rua.

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

Tangohia te 2 i te 6, ka 4.

$$x-2y=4$$

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 2,3.

$$3x+2\times 2y=6x-8$$

Whakareatia te 2 ki te 2, ka 4.

$$3x+4y=6x-8$$

Tangohia te 6x mai i ngā taha e rua.

$$3x+4y-6x=-8$$

Pahekotia te 3x me -6x, ka -3x.

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

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.

$$x-2y=4,-3x+4y=-8$$

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

$$x-2y=4$$

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

$$x=2y+4$$

Whakakapia te 4+2y mō te x ki tērā atu whārite, -3x+4y=-8.

$$-3\left(2y+4\right)+4y=-8$$

Whakareatia -3 ki te 4+2y.

$$-6y-12+4y=-8$$

Tāpiri -6y ki te 4y.

$$-2y-12=-8$$

Me tāpiri 12 ki ngā taha e rua o te whārite.

$$-2y=4$$

Whakawehea ngā taha e rua ki te -2.

$$y=-2$$

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

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

Whakareatia 2 ki te -2.

$$x=-4+4$$

Tāpiri 4 ki te -4.

$$x=0$$

Kua oti te pūnaha te whakatau.

$$x=0,y=-2$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Whakareatia ngā taha e rua o te whārite ki te 3.

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

Pahekotia te y me -3y, ka -2y.

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

Tangohia te 2 mai i ngā taha e rua.

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

Tangohia te 2 i te 6, ka 4.

$$x-2y=4$$

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 2,3.

$$3x+2\times 2y=6x-8$$

Whakareatia te 2 ki te 2, ka 4.

$$3x+4y=6x-8$$

Tangohia te 6x mai i ngā taha e rua.

$$3x+4y-6x=-8$$

Pahekotia te 3x me -6x, ka -3x.

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

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

$$x-2y=4,-3x+4y=-8$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-2\\-3&4\end{matrix}\right).

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}0\\-2\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=0,y=-2$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Whakareatia ngā taha e rua o te whārite ki te 3.

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

Pahekotia te y me -3y, ka -2y.

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

Tangohia te 2 mai i ngā taha e rua.

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

Tangohia te 2 i te 6, ka 4.

$$x-2y=4$$

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 2,3.

$$3x+2\times 2y=6x-8$$

Whakareatia te 2 ki te 2, ka 4.

$$3x+4y=6x-8$$

Tangohia te 6x mai i ngā taha e rua.

$$3x+4y-6x=-8$$

Pahekotia te 3x me -6x, ka -3x.

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

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.

$$x-2y=4,-3x+4y=-8$$

Kia ōrite ai a x 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 1.

$$-3x-3\left(-2\right)y=-3\times 4,-3x+4y=-8$$

Whakarūnātia.

$$-3x+6y=-12,-3x+4y=-8$$

Me tango -3x+4y=-8 mai i -3x+6y=-12 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-3x+3x+6y-4y=-12+8$$

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.

$$6y-4y=-12+8$$

Tāpiri 6y ki te -4y.

$$2y=-12+8$$

Tāpiri -12 ki te 8.

$$2y=-4$$

Whakawehea ngā taha e rua ki te 2.

$$y=-2$$

Whakaurua te -2 mō y ki -3x+4y=-8. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-3x+4\left(-2\right)=-8$$

Whakareatia 4 ki te -2.

$$-3x-8=-8$$

Me tāpiri 8 ki 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.

$$x=0,y=-2$$