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

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

Whakaoti mō x, y

$$x=12$$

$$y=-2$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te -2 ki te x+y.

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

Pahekotia te x me -2x, ka -x.

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

Tangohia te 3y mai i ngā taha e rua.

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

Pahekotia te -2y me -3y, ka -5y.

$$-x-5y=-2$$

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 3,2.

$$2x+3y=18$$

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.

$$-x-5y=-2,2x+3y=18$$

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

$$-x-5y=-2$$

Me tāpiri 5y ki ngā taha e rua o te whārite.

$$-x=5y-2$$

Whakawehea ngā taha e rua ki te -1.

$$x=-\left(5y-2\right)$$

Whakareatia -1 ki te 5y-2.

$$x=-5y+2$$

Whakakapia te -5y+2 mō te x ki tērā atu whārite, 2x+3y=18.

$$2\left(-5y+2\right)+3y=18$$

Whakareatia 2 ki te -5y+2.

$$-10y+4+3y=18$$

Tāpiri -10y ki te 3y.

$$-7y+4=18$$

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

$$-7y=14$$

Whakawehea ngā taha e rua ki te -7.

$$y=-2$$

Whakaurua te -2 mō y ki x=-5y+2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-5\left(-2\right)+2$$

Whakareatia -5 ki te -2.

$$x=10+2$$

Tāpiri 2 ki te 10.

$$x=12$$

Kua oti te pūnaha te whakatau.

$$x=12,y=-2$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te -2 ki te x+y.

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

Pahekotia te x me -2x, ka -x.

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

Tangohia te 3y mai i ngā taha e rua.

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

Pahekotia te -2y me -3y, ka -5y.

$$-x-5y=-2$$

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 3,2.

$$2x+3y=18$$

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

$$-x-5y=-2,2x+3y=18$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-1&-5\\2&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-2\\18\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-1&-5\\2&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}-1&-5\\2&3\end{matrix}\right))\left(\begin{matrix}-1&-5\\2&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&-5\\2&3\end{matrix}\right))\left(\begin{matrix}-2\\18\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&-5\\2&3\end{matrix}\right))\left(\begin{matrix}-2\\18\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&-5\\2&3\end{matrix}\right))\left(\begin{matrix}-2\\18\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{-3-\left(-5\times 2\right)}&-\frac{-5}{-3-\left(-5\times 2\right)}\\-\frac{2}{-3-\left(-5\times 2\right)}&-\frac{1}{-3-\left(-5\times 2\right)}\end{matrix}\right)\left(\begin{matrix}-2\\18\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{7}&\frac{5}{7}\\-\frac{2}{7}&-\frac{1}{7}\end{matrix}\right)\left(\begin{matrix}-2\\18\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{7}\left(-2\right)+\frac{5}{7}\times 18\\-\frac{2}{7}\left(-2\right)-\frac{1}{7}\times 18\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}12\\-2\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=12,y=-2$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te -2 ki te x+y.

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

Pahekotia te x me -2x, ka -x.

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

Tangohia te 3y mai i ngā taha e rua.

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

Pahekotia te -2y me -3y, ka -5y.

$$-x-5y=-2$$

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 3,2.

$$2x+3y=18$$

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.

$$-x-5y=-2,2x+3y=18$$

Kia ōrite ai a -x 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 -1.

$$2\left(-1\right)x+2\left(-5\right)y=2\left(-2\right),-2x-3y=-18$$

Whakarūnātia.

$$-2x-10y=-4,-2x-3y=-18$$

Me tango -2x-3y=-18 mai i -2x-10y=-4 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-2x+2x-10y+3y=-4+18$$

Tāpiri -2x ki te 2x. Ka whakakore atu ngā kupu -2x me 2x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-10y+3y=-4+18$$

Tāpiri -10y ki te 3y.

$$-7y=-4+18$$

Tāpiri -4 ki te 18.

$$-7y=14$$

Whakawehea ngā taha e rua ki te -7.

$$y=-2$$

Whakaurua te -2 mō y ki 2x+3y=18. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x+3\left(-2\right)=18$$

Whakareatia 3 ki te -2.

$$2x-6=18$$

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

$$2x=24$$

Whakawehea ngā taha e rua ki te 2.

$$x=12$$

Kua oti te pūnaha te whakatau.

$$x=12,y=-2$$