Whakaoti i te {l}{6x+6y=6}{6x+3y=-3}

$$\left\{ \begin{array} { l } { 6 x + 6 y = 6 } \\ { 6 x + 3 y = - 3 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-2$$

$$y=3$$

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.

$$6x+6y=6,6x+3y=-3$$

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

$$6x+6y=6$$

Me tango 6y mai i ngā taha e rua o te whārite.

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

Whakawehea ngā taha e rua ki te 6.

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

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

$$x=-y+1$$

Whakakapia te -y+1 mō te x ki tērā atu whārite, 6x+3y=-3.

$$6\left(-y+1\right)+3y=-3$$

Whakareatia 6 ki te -y+1.

$$-6y+6+3y=-3$$

Tāpiri -6y ki te 3y.

$$-3y+6=-3$$

Me tango 6 mai i ngā taha e rua o te whārite.

$$-3y=-9$$

Whakawehea ngā taha e rua ki te -3.

$$y=3$$

Whakaurua te 3 mō y ki x=-y+1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-3+1$$

Tāpiri 1 ki te -3.

$$x=-2$$

Kua oti te pūnaha te whakatau.

$$x=-2,y=3$$

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.

$$6x+6y=6,6x+3y=-3$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=-2,y=3$$

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.

$$6x+6y=6,6x+3y=-3$$

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

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

Tāpiri 6x ki te -6x. Ka whakakore atu ngā kupu 6x me -6x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$6y-3y=6+3$$

Tāpiri 6y ki te -3y.

$$3y=6+3$$

Tāpiri 6 ki te 3.

$$3y=9$$

Whakawehea ngā taha e rua ki te 3.

$$y=3$$

Whakaurua te 3 mō y ki 6x+3y=-3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$6x+3\times 3=-3$$

Whakareatia 3 ki te 3.

$$6x+9=-3$$

Me tango 9 mai i ngā taha e rua o te whārite.

$$6x=-12$$

Whakawehea ngā taha e rua ki te 6.

$$x=-2$$

Kua oti te pūnaha te whakatau.

$$x=-2,y=3$$