Whakaoti i te {c}{x-3-2(y+1)=-12}{3(x-2y)-2y=-21}

$$\left\{ \begin{array} { c } { x - 3 - 2 ( y + 1 ) = - 12 } \\ { 3 ( x - 2 y ) - 2 y = - 21 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-7$$

$$y=0$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te -2 ki te y+1.

$$x-3-2y-2=-12$$

Tangohia te 2 i te -3, ka -5.

$$x-5-2y=-12$$

Me tāpiri te 5 ki ngā taha e rua.

$$x-2y=-12+5$$

Tāpirihia te -12 ki te 5, ka -7.

$$x-2y=-7$$

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

$$3x-6y-2y=-21$$

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

$$3x-8y=-21$$

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.

$$x-2y=-7,3x-8y=-21$$

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

$$x-2y=-7$$

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

$$x=2y-7$$

Whakakapia te 2y-7 mō te x ki tērā atu whārite, 3x-8y=-21.

$$3\left(2y-7\right)-8y=-21$$

Whakareatia 3 ki te 2y-7.

$$6y-21-8y=-21$$

Tāpiri 6y ki te -8y.

$$-2y-21=-21$$

Me tāpiri 21 ki ngā taha e rua o te whārite.

$$-2y=0$$

Whakawehea ngā taha e rua ki te -2.

$$y=0$$

Whakaurua te 0 mō y ki x=2y-7. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-7$$

Kua oti te pūnaha te whakatau.

$$x=-7,y=0$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te -2 ki te y+1.

$$x-3-2y-2=-12$$

Tangohia te 2 i te -3, ka -5.

$$x-5-2y=-12$$

Me tāpiri te 5 ki ngā taha e rua.

$$x-2y=-12+5$$

Tāpirihia te -12 ki te 5, ka -7.

$$x-2y=-7$$

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

$$3x-6y-2y=-21$$

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

$$3x-8y=-21$$

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

$$x-2y=-7,3x-8y=-21$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-2\\3&-8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-7\\-21\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-2\\3&-8\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-2\\3&-8\end{matrix}\right))\left(\begin{matrix}1&-2\\3&-8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\3&-8\end{matrix}\right))\left(\begin{matrix}-7\\-21\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\3&-8\end{matrix}\right))\left(\begin{matrix}-7\\-21\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\3&-8\end{matrix}\right))\left(\begin{matrix}-7\\-21\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{8}{-8-\left(-2\times 3\right)}&-\frac{-2}{-8-\left(-2\times 3\right)}\\-\frac{3}{-8-\left(-2\times 3\right)}&\frac{1}{-8-\left(-2\times 3\right)}\end{matrix}\right)\left(\begin{matrix}-7\\-21\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4&-1\\\frac{3}{2}&-\frac{1}{2}\end{matrix}\right)\left(\begin{matrix}-7\\-21\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\left(-7\right)-\left(-21\right)\\\frac{3}{2}\left(-7\right)-\frac{1}{2}\left(-21\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-7\\0\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-7,y=0$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te -2 ki te y+1.

$$x-3-2y-2=-12$$

Tangohia te 2 i te -3, ka -5.

$$x-5-2y=-12$$

Me tāpiri te 5 ki ngā taha e rua.

$$x-2y=-12+5$$

Tāpirihia te -12 ki te 5, ka -7.

$$x-2y=-7$$

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

$$3x-6y-2y=-21$$

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

$$3x-8y=-21$$

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.

$$x-2y=-7,3x-8y=-21$$

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.

$$3x+3\left(-2\right)y=3\left(-7\right),3x-8y=-21$$

Whakarūnātia.

$$3x-6y=-21,3x-8y=-21$$

Me tango 3x-8y=-21 mai i 3x-6y=-21 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3x-3x-6y+8y=-21+21$$

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.

$$-6y+8y=-21+21$$

Tāpiri -6y ki te 8y.

$$2y=-21+21$$

Tāpiri -21 ki te 21.

$$2y=0$$

Whakawehea ngā taha e rua ki te 2.

$$y=0$$

Whakaurua te 0 mō y ki 3x-8y=-21. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x=-21$$

Whakawehea ngā taha e rua ki te 3.

$$x=-7$$

Kua oti te pūnaha te whakatau.

$$x=-7,y=0$$