Whakaoti i te {l}{6x+12y=-6}{2x+5y=0}

$$\left. \begin{array} { l } { 6 x + 12 y = - 6 } \\ { 2 x + 5 y = 0 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-5$$

$$y=2$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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.

$$6x+12y=-6,2x+5y=0$$

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

$$6x+12y=-6$$

Me tango 12y mai i ngā taha e rua o te whārite.

$$6x=-12y-6$$

Whakawehea ngā taha e rua ki te 6.

$$x=\frac{1}{6}\left(-12y-6\right)$$

Whakareatia \frac{1}{6} ki te -12y-6.

$$x=-2y-1$$

Whakakapia te -2y-1 mō te x ki tērā atu whārite, 2x+5y=0.

$$2\left(-2y-1\right)+5y=0$$

Whakareatia 2 ki te -2y-1.

$$-4y-2+5y=0$$

Tāpiri -4y ki te 5y.

$$y-2=0$$

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

$$y=2$$

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

$$x=-2\times 2-1$$

Whakareatia -2 ki te 2.

$$x=-4-1$$

Tāpiri -1 ki te -4.

$$x=-5$$

Kua oti te pūnaha te whakatau.

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

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$6x+12y=-6,2x+5y=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}6&12\\2&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-6\\0\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}6&12\\2&5\end{matrix}\right).

$$inverse(\left(\begin{matrix}6&12\\2&5\end{matrix}\right))\left(\begin{matrix}6&12\\2&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&12\\2&5\end{matrix}\right))\left(\begin{matrix}-6\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&12\\2&5\end{matrix}\right))\left(\begin{matrix}-6\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&12\\2&5\end{matrix}\right))\left(\begin{matrix}-6\\0\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{6\times 5-12\times 2}&-\frac{12}{6\times 5-12\times 2}\\-\frac{2}{6\times 5-12\times 2}&\frac{6}{6\times 5-12\times 2}\end{matrix}\right)\left(\begin{matrix}-6\\0\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{6}&-2\\-\frac{1}{3}&1\end{matrix}\right)\left(\begin{matrix}-6\\0\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{6}\left(-6\right)\\-\frac{1}{3}\left(-6\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

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

Ngā Upane e Whakamahi i te Tangohanga

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.

$$6x+12y=-6,2x+5y=0$$

Kia ōrite ai a 6x 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 6.

$$2\times 6x+2\times 12y=2\left(-6\right),6\times 2x+6\times 5y=0$$

Whakarūnātia.

$$12x+24y=-12,12x+30y=0$$

Me tango 12x+30y=0 mai i 12x+24y=-12 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$12x-12x+24y-30y=-12$$

Tāpiri 12x ki te -12x. Ka whakakore atu ngā kupu 12x me -12x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$24y-30y=-12$$

Tāpiri 24y ki te -30y.

$$-6y=-12$$

Whakawehea ngā taha e rua ki te -6.

$$y=2$$

Whakaurua te 2 mō y ki 2x+5y=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x+5\times 2=0$$

Whakareatia 5 ki te 2.

$$2x+10=0$$

Me tango 10 mai i ngā taha e rua o te whārite.

$$2x=-10$$

Whakawehea ngā taha e rua ki te 2.

$$x=-5$$

Kua oti te pūnaha te whakatau.

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