Whakaoti i te {l}{6x+2y=-6}{3x-y=9}

$$\left. \begin{array} { l } { 6 x + 2 y = - 6 } \\ { 3 x - y = 9 } \end{array} \right.$$

Whakaoti mō x, y

$$x=1$$

$$y=-6$$

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.

$$6x+2y=-6,3x-y=9$$

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

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

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

$$6x=-2y-6$$

Whakawehea ngā taha e rua ki te 6.

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

Whakareatia \frac{1}{6} ki te -2y-6.

$$x=-\frac{1}{3}y-1$$

Whakakapia te -\frac{y}{3}-1 mō te x ki tērā atu whārite, 3x-y=9.

$$3\left(-\frac{1}{3}y-1\right)-y=9$$

Whakareatia 3 ki te -\frac{y}{3}-1.

$$-y-3-y=9$$

Tāpiri -y ki te -y.

$$-2y-3=9$$

Me tāpiri 3 ki ngā taha e rua o te whārite.

$$-2y=12$$

Whakawehea ngā taha e rua ki te -2.

$$y=-6$$

Whakaurua te -6 mō y ki x=-\frac{1}{3}y-1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{3}\left(-6\right)-1$$

Whakareatia -\frac{1}{3} ki te -6.

$$x=2-1$$

Tāpiri -1 ki te 2.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=-6$$

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.

$$6x+2y=-6,3x-y=9$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}6&2\\3&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-6\\9\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}6&2\\3&-1\end{matrix}\right))\left(\begin{matrix}6&2\\3&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&2\\3&-1\end{matrix}\right))\left(\begin{matrix}-6\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&2\\3&-1\end{matrix}\right))\left(\begin{matrix}-6\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&2\\3&-1\end{matrix}\right))\left(\begin{matrix}-6\\9\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{6\left(-1\right)-2\times 3}&-\frac{2}{6\left(-1\right)-2\times 3}\\-\frac{3}{6\left(-1\right)-2\times 3}&\frac{6}{6\left(-1\right)-2\times 3}\end{matrix}\right)\left(\begin{matrix}-6\\9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{12}&\frac{1}{6}\\\frac{1}{4}&-\frac{1}{2}\end{matrix}\right)\left(\begin{matrix}-6\\9\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{12}\left(-6\right)+\frac{1}{6}\times 9\\\frac{1}{4}\left(-6\right)-\frac{1}{2}\times 9\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=1,y=-6$$

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.

$$6x+2y=-6,3x-y=9$$

Kia ōrite ai a 6x 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 6.

$$3\times 6x+3\times 2y=3\left(-6\right),6\times 3x+6\left(-1\right)y=6\times 9$$

Whakarūnātia.

$$18x+6y=-18,18x-6y=54$$

Me tango 18x-6y=54 mai i 18x+6y=-18 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$18x-18x+6y+6y=-18-54$$

Tāpiri 18x ki te -18x. Ka whakakore atu ngā kupu 18x me -18x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$6y+6y=-18-54$$

Tāpiri 6y ki te 6y.

$$12y=-18-54$$

Tāpiri -18 ki te -54.

$$12y=-72$$

Whakawehea ngā taha e rua ki te 12.

$$y=-6$$

Whakaurua te -6 mō y ki 3x-y=9. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x-\left(-6\right)=9$$

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

$$3x=3$$

Whakawehea ngā taha e rua ki te 3.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=-6$$