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

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

Whakaoti mō x, y

$$x=-2$$

$$y=4$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Hei kimi i te tauaro o x+2y, kimihia te tauaro o ia taurangi.

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

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

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

Me tāpiri te y ki ngā taha e rua.

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

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

$$-2x-y=0$$

Whakaarohia te whārite tuarua. Tangohia te 2y mai i ngā taha e rua.

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

Me tāpiri te x ki ngā taha e rua.

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

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

$$-2x-2y=-4$$

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.

$$-2x-y=0,-2x-2y=-4$$

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

$$-2x-y=0$$

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

$$-2x=y$$

Whakawehea ngā taha e rua ki te -2.

$$x=-\frac{1}{2}y$$

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

$$-2\left(-\frac{1}{2}\right)y-2y=-4$$

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

$$y-2y=-4$$

Tāpiri y ki te -2y.

$$-y=-4$$

Whakawehea ngā taha e rua ki te -1.

$$y=4$$

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

$$x=-\frac{1}{2}\times 4$$

Whakareatia -\frac{1}{2} ki te 4.

$$x=-2$$

Kua oti te pūnaha te whakatau.

$$x=-2,y=4$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Hei kimi i te tauaro o x+2y, kimihia te tauaro o ia taurangi.

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

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

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

Me tāpiri te y ki ngā taha e rua.

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

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

$$-2x-y=0$$

Whakaarohia te whārite tuarua. Tangohia te 2y mai i ngā taha e rua.

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

Me tāpiri te x ki ngā taha e rua.

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

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

$$-2x-2y=-4$$

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

$$-2x-y=0,-2x-2y=-4$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=-2,y=4$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Hei kimi i te tauaro o x+2y, kimihia te tauaro o ia taurangi.

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

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

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

Me tāpiri te y ki ngā taha e rua.

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

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

$$-2x-y=0$$

Whakaarohia te whārite tuarua. Tangohia te 2y mai i ngā taha e rua.

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

Me tāpiri te x ki ngā taha e rua.

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

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

$$-2x-2y=-4$$

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.

$$-2x-y=0,-2x-2y=-4$$

Me tango -2x-2y=-4 mai i -2x-y=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-2x+2x-y+2y=4$$

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.

$$-y+2y=4$$

Tāpiri -y ki te 2y.

$$y=4$$

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

$$-2x-2\times 4=-4$$

Whakareatia -2 ki te 4.

$$-2x-8=-4$$

Me tāpiri 8 ki ngā taha e rua o te whārite.

$$-2x=4$$

Whakawehea ngā taha e rua ki te -2.

$$x=-2$$

Kua oti te pūnaha te whakatau.

$$x=-2,y=4$$