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

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

Whakaoti mō x, y

$$x=0$$

$$y=0$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te -3 ki te 3x-y.

$$-9x+3y=2\left(y+x\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te y+x.

$$-9x+3y=2y+2x$$

Tangohia te 2y mai i ngā taha e rua.

$$-9x+3y-2y=2x$$

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

$$-9x+y=2x$$

Tangohia te 2x mai i ngā taha e rua.

$$-9x+y-2x=0$$

Pahekotia te -9x me -2x, ka -11x.

$$-11x+y=0$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te -3 ki te 2x+y.

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

Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te x-3y.

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

Tangohia te 2x mai i ngā taha e rua.

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

Pahekotia te -6x me -2x, ka -8x.

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

Me tāpiri te 6y ki ngā taha e rua.

$$-8x-3y+6y=0$$

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

$$-8x+3y=0$$

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.

$$-11x+y=0,-8x+3y=0$$

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

$$-11x+y=0$$

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

$$-11x=-y$$

Whakawehea ngā taha e rua ki te -11.

$$x=-\frac{1}{11}\left(-1\right)y$$

Whakareatia -\frac{1}{11} ki te -y.

$$x=\frac{1}{11}y$$

Whakakapia te \frac{y}{11} mō te x ki tērā atu whārite, -8x+3y=0.

$$-8\times \frac{1}{11}y+3y=0$$

Whakareatia -8 ki te \frac{y}{11}.

$$-\frac{8}{11}y+3y=0$$

Tāpiri -\frac{8y}{11} ki te 3y.

$$\frac{25}{11}y=0$$

Whakawehea ngā taha e rua o te whārite ki te \frac{25}{11}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=0$$

Whakaurua te 0 mō y ki x=\frac{1}{11}y. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=0$$

Kua oti te pūnaha te whakatau.

$$x=0,y=0$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te -3 ki te 3x-y.

$$-9x+3y=2\left(y+x\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te y+x.

$$-9x+3y=2y+2x$$

Tangohia te 2y mai i ngā taha e rua.

$$-9x+3y-2y=2x$$

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

$$-9x+y=2x$$

Tangohia te 2x mai i ngā taha e rua.

$$-9x+y-2x=0$$

Pahekotia te -9x me -2x, ka -11x.

$$-11x+y=0$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te -3 ki te 2x+y.

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

Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te x-3y.

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

Tangohia te 2x mai i ngā taha e rua.

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

Pahekotia te -6x me -2x, ka -8x.

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

Me tāpiri te 6y ki ngā taha e rua.

$$-8x-3y+6y=0$$

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

$$-8x+3y=0$$

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

$$-11x+y=0,-8x+3y=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-11&1\\-8&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}0\\0\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-11&1\\-8&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}-11&1\\-8&3\end{matrix}\right))\left(\begin{matrix}-11&1\\-8&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-11&1\\-8&3\end{matrix}\right))\left(\begin{matrix}0\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-11&1\\-8&3\end{matrix}\right))\left(\begin{matrix}0\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-11&1\\-8&3\end{matrix}\right))\left(\begin{matrix}0\\0\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{-11\times 3-\left(-8\right)}&-\frac{1}{-11\times 3-\left(-8\right)}\\-\frac{-8}{-11\times 3-\left(-8\right)}&-\frac{11}{-11\times 3-\left(-8\right)}\end{matrix}\right)\left(\begin{matrix}0\\0\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{25}&\frac{1}{25}\\-\frac{8}{25}&\frac{11}{25}\end{matrix}\right)\left(\begin{matrix}0\\0\end{matrix}\right)$$

Whakareatia ngā poukapa.

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

Tangohia ngā huānga poukapa x me y.

$$x=0,y=0$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te -3 ki te 3x-y.

$$-9x+3y=2\left(y+x\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te y+x.

$$-9x+3y=2y+2x$$

Tangohia te 2y mai i ngā taha e rua.

$$-9x+3y-2y=2x$$

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

$$-9x+y=2x$$

Tangohia te 2x mai i ngā taha e rua.

$$-9x+y-2x=0$$

Pahekotia te -9x me -2x, ka -11x.

$$-11x+y=0$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te -3 ki te 2x+y.

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

Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te x-3y.

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

Tangohia te 2x mai i ngā taha e rua.

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

Pahekotia te -6x me -2x, ka -8x.

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

Me tāpiri te 6y ki ngā taha e rua.

$$-8x-3y+6y=0$$

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

$$-8x+3y=0$$

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.

$$-11x+y=0,-8x+3y=0$$

Kia ōrite ai a -11x me -8x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -8 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te -11.

$$-8\left(-11\right)x-8y=0,-11\left(-8\right)x-11\times 3y=0$$

Whakarūnātia.

$$88x-8y=0,88x-33y=0$$

Me tango 88x-33y=0 mai i 88x-8y=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$88x-88x-8y+33y=0$$

Tāpiri 88x ki te -88x. Ka whakakore atu ngā kupu 88x me -88x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-8y+33y=0$$

Tāpiri -8y ki te 33y.

$$25y=0$$

Whakawehea ngā taha e rua ki te 25.

$$y=0$$

Whakaurua te 0 mō y ki -8x+3y=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-8x=0$$

Whakawehea ngā taha e rua ki te -8.

$$x=0$$

Kua oti te pūnaha te whakatau.

$$x=0,y=0$$