Whakaoti i te {l}{x-y=2x}{2x+y=16}

$$\left\{ \begin{array} { l } { x - y = 2 x } \\ { 2 x + y = 16 } \end{array} \right.$$

Whakaoti mō x, y

$$x=16$$

$$y=-16$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tangohia te 2x mai i ngā taha e rua.

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

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

$$-x-y=0$$

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.

$$-x-y=0,2x+y=16$$

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

$$-x-y=0$$

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

$$-x=y$$

Whakawehea ngā taha e rua ki te -1.

$$x=-y$$

Whakakapia te -y mō te x ki tērā atu whārite, 2x+y=16.

$$2\left(-1\right)y+y=16$$

Whakareatia 2 ki te -y.

$$-2y+y=16$$

Tāpiri -2y ki te y.

$$-y=16$$

Whakawehea ngā taha e rua ki te -1.

$$y=-16$$

Whakaurua te -16 mō y ki x=-y. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\left(-16\right)$$

Whakareatia -1 ki te -16.

$$x=16$$

Kua oti te pūnaha te whakatau.

$$x=16,y=-16$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tangohia te 2x mai i ngā taha e rua.

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

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

$$-x-y=0$$

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

$$-x-y=0,2x+y=16$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-1&-1\\2&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}0\\16\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}-1&-1\\2&1\end{matrix}\right))\left(\begin{matrix}-1&-1\\2&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&-1\\2&1\end{matrix}\right))\left(\begin{matrix}0\\16\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&-1\\2&1\end{matrix}\right))\left(\begin{matrix}0\\16\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&-1\\2&1\end{matrix}\right))\left(\begin{matrix}0\\16\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{-1-\left(-2\right)}&-\frac{-1}{-1-\left(-2\right)}\\-\frac{2}{-1-\left(-2\right)}&-\frac{1}{-1-\left(-2\right)}\end{matrix}\right)\left(\begin{matrix}0\\16\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1&1\\-2&-1\end{matrix}\right)\left(\begin{matrix}0\\16\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}16\\-16\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=16,y=-16$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Tangohia te 2x mai i ngā taha e rua.

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

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

$$-x-y=0$$

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.

$$-x-y=0,2x+y=16$$

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.

$$2\left(-1\right)x+2\left(-1\right)y=0,-2x-y=-16$$

Whakarūnātia.

$$-2x-2y=0,-2x-y=-16$$

Me tango -2x-y=-16 mai i -2x-2y=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-2x+2x-2y+y=16$$

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.

$$-2y+y=16$$

Tāpiri -2y ki te y.

$$-y=16$$

Whakawehea ngā taha e rua ki te -1.

$$y=-16$$

Whakaurua te -16 mō y ki 2x+y=16. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x-16=16$$

Me tāpiri 16 ki ngā taha e rua o te whārite.

$$2x=32$$

Whakawehea ngā taha e rua ki te 2.

$$x=16$$

Kua oti te pūnaha te whakatau.

$$x=16,y=-16$$