Whakaoti i te {l}{6x+2y=20}{-4x+y=-11}

$$\left. \begin{array} { l } { 6 x + 2 y = 20 } \\ { - 4 x + y = - 11 } \end{array} \right.$$

Whakaoti mō x, y

$$x=3$$

$$y=1$$

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.

$$6x+2y=20,-4x+y=-11$$

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

$$6x+2y=20$$

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

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

Whakawehea ngā taha e rua ki te 6.

$$x=\frac{1}{6}\left(-2y+20\right)$$

Whakareatia \frac{1}{6} ki te -2y+20.

$$x=-\frac{1}{3}y+\frac{10}{3}$$

Whakakapia te \frac{-y+10}{3} mō te x ki tērā atu whārite, -4x+y=-11.

$$-4\left(-\frac{1}{3}y+\frac{10}{3}\right)+y=-11$$

Whakareatia -4 ki te \frac{-y+10}{3}.

$$\frac{4}{3}y-\frac{40}{3}+y=-11$$

Tāpiri \frac{4y}{3} ki te y.

$$\frac{7}{3}y-\frac{40}{3}=-11$$

Me tāpiri \frac{40}{3} ki ngā taha e rua o te whārite.

$$\frac{7}{3}y=\frac{7}{3}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{7}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=1$$

Whakaurua te 1 mō y ki x=-\frac{1}{3}y+\frac{10}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{-1+10}{3}$$

Tāpiri \frac{10}{3} ki te -\frac{1}{3} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=1$$

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.

$$6x+2y=20,-4x+y=-11$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}6&2\\-4&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}20\\-11\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}6&2\\-4&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}6&2\\-4&1\end{matrix}\right))\left(\begin{matrix}6&2\\-4&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&2\\-4&1\end{matrix}\right))\left(\begin{matrix}20\\-11\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&2\\-4&1\end{matrix}\right))\left(\begin{matrix}20\\-11\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&2\\-4&1\end{matrix}\right))\left(\begin{matrix}20\\-11\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{6-2\left(-4\right)}&-\frac{2}{6-2\left(-4\right)}\\-\frac{-4}{6-2\left(-4\right)}&\frac{6}{6-2\left(-4\right)}\end{matrix}\right)\left(\begin{matrix}20\\-11\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{14}&-\frac{1}{7}\\\frac{2}{7}&\frac{3}{7}\end{matrix}\right)\left(\begin{matrix}20\\-11\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{14}\times 20-\frac{1}{7}\left(-11\right)\\\frac{2}{7}\times 20+\frac{3}{7}\left(-11\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=3,y=1$$

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.

$$6x+2y=20,-4x+y=-11$$

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.

$$-4\times 6x-4\times 2y=-4\times 20,6\left(-4\right)x+6y=6\left(-11\right)$$

Whakarūnātia.

$$-24x-8y=-80,-24x+6y=-66$$

Me tango -24x+6y=-66 mai i -24x-8y=-80 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-24x+24x-8y-6y=-80+66$$

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.

$$-8y-6y=-80+66$$

Tāpiri -8y ki te -6y.

$$-14y=-80+66$$

Tāpiri -80 ki te 66.

$$-14y=-14$$

Whakawehea ngā taha e rua ki te -14.

$$y=1$$

Whakaurua te 1 mō y ki -4x+y=-11. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-4x+1=-11$$

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

$$-4x=-12$$

Whakawehea ngā taha e rua ki te -4.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=1$$