Whakaoti i te {l}{-6x+2y=6}{4x-4y=-20}

$$\left. \begin{array} { l } { - 6 x + 2 y = 6 } \\ { 4 x - 4 y = - 20 } \end{array} \right.$$

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.

$$-6x+2y=6,4x-4y=-20$$

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, 4x-4y=-20.

$$4\left(\frac{1}{3}y-1\right)-4y=-20$$

Whakareatia 4 ki te \frac{y}{3}-1.

$$\frac{4}{3}y-4-4y=-20$$

Tāpiri \frac{4y}{3} ki te -4y.

$$-\frac{8}{3}y-4=-20$$

Me tāpiri 4 ki ngā taha e rua o te whārite.

$$-\frac{8}{3}y=-16$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{8}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=6$$

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.

$$x=\frac{1}{3}\times 6-1$$

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.

$$-6x+2y=6,4x-4y=-20$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-6&2\\4&-4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}6\\-20\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-6&2\\4&-4\end{matrix}\right).

$$inverse(\left(\begin{matrix}-6&2\\4&-4\end{matrix}\right))\left(\begin{matrix}-6&2\\4&-4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-6&2\\4&-4\end{matrix}\right))\left(\begin{matrix}6\\-20\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-6&2\\4&-4\end{matrix}\right))\left(\begin{matrix}6\\-20\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-6&2\\4&-4\end{matrix}\right))\left(\begin{matrix}6\\-20\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{-6\left(-4\right)-2\times 4}&-\frac{2}{-6\left(-4\right)-2\times 4}\\-\frac{4}{-6\left(-4\right)-2\times 4}&-\frac{6}{-6\left(-4\right)-2\times 4}\end{matrix}\right)\left(\begin{matrix}6\\-20\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{4}&-\frac{1}{8}\\-\frac{1}{4}&-\frac{3}{8}\end{matrix}\right)\left(\begin{matrix}6\\-20\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{4}\times 6-\frac{1}{8}\left(-20\right)\\-\frac{1}{4}\times 6-\frac{3}{8}\left(-20\right)\end{matrix}\right)$$

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.

$$-6x+2y=6,4x-4y=-20$$

Kia ōrite ai a -6x me 4x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 4 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te -6.

$$4\left(-6\right)x+4\times 2y=4\times 6,-6\times 4x-6\left(-4\right)y=-6\left(-20\right)$$

Whakarūnātia.

$$-24x+8y=24,-24x+24y=120$$

Me tango -24x+24y=120 mai i -24x+8y=24 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-24x+24x+8y-24y=24-120$$

Tāpiri -24x ki te 24x. Ka whakakore atu ngā kupu -24x me 24x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$8y-24y=24-120$$

Tāpiri 8y ki te -24y.

$$-16y=24-120$$

Tāpiri 24 ki te -120.

$$-16y=-96$$

Whakawehea ngā taha e rua ki te -16.

$$y=6$$

Whakaurua te 6 mō y ki 4x-4y=-20. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x-4\times 6=-20$$

Whakareatia -4 ki te 6.

$$4x-24=-20$$

Me tāpiri 24 ki ngā taha e rua o te whārite.

$$4x=4$$

Whakawehea ngā taha e rua ki te 4.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=6$$