Whakaoti i te {l}{x-2(3y-1)=-4}{2/3y-(-x-7)=1}

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

Whakaoti mō x, y

$$x=-6$$

$$y=0$$

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.

$$x-2\left(3y-1\right)=-4,-\left(-x-7\right)+\frac{2}{3}y=1$$

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

$$x-2\left(3y-1\right)=-4$$

Whakareatia -2 ki te 3y-1.

$$x-6y+2=-4$$

Me tango 2 mai i ngā taha e rua o te whārite.

$$x-6y=-6$$

Me tāpiri 6y ki ngā taha e rua o te whārite.

$$x=6y-6$$

Whakakapia te -6+6y mō te x ki tērā atu whārite, -\left(-x-7\right)+\frac{2}{3}y=1.

$$-\left(-\left(6y-6\right)-7\right)+\frac{2}{3}y=1$$

Whakareatia -1 ki te -6+6y.

$$-\left(-6y+6-7\right)+\frac{2}{3}y=1$$

Tāpiri 6 ki te -7.

$$-\left(-6y-1\right)+\frac{2}{3}y=1$$

Whakareatia -1 ki te -6y-1.

$$6y+1+\frac{2}{3}y=1$$

Tāpiri 6y ki te \frac{2y}{3}.

$$\frac{20}{3}y+1=1$$

Me tango 1 mai i ngā taha e rua o te whārite.

$$\frac{20}{3}y=0$$

Whakawehea ngā taha e rua o te whārite ki te \frac{20}{3}, 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=6y-6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-6$$

Kua oti te pūnaha te whakatau.

$$x=-6,y=0$$

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.

$$x-2\left(3y-1\right)=-4,-\left(-x-7\right)+\frac{2}{3}y=1$$

Whakarūnātia te whārite tuatahi ki te āhua tānga ngahuru.

$$x-2\left(3y-1\right)=-4$$

Whakareatia -2 ki te 3y-1.

$$x-6y+2=-4$$

Me tango 2 mai i ngā taha e rua o te whārite.

$$x-6y=-6$$

Whakarūnātia te whārite tuarua ki te āhua tānga ngahuru.

$$-\left(-x-7\right)+\frac{2}{3}y=1$$

Whakareatia -1 ki te -x-7.

$$x+7+\frac{2}{3}y=1$$

Me tango 7 mai i ngā taha e rua o te whārite.

$$x+\frac{2}{3}y=-6$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-6\\1&\frac{2}{3}\end{matrix}\right).

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{10}&\frac{9}{10}\\-\frac{3}{20}&\frac{3}{20}\end{matrix}\right)\left(\begin{matrix}-6\\-6\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{10}\left(-6\right)+\frac{9}{10}\left(-6\right)\\-\frac{3}{20}\left(-6\right)+\frac{3}{20}\left(-6\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=-6,y=0$$