Whakaoti i te {l}{2x+2y=6}{7y-5x=11}

$$\left. \begin{array} { l } { 2 x + 2 y = 6 } \\ { 7 y - 5 x = 11 } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{5}{6}\approx 0.833333333$$

$$y = \frac{13}{6} = 2\frac{1}{6} \approx 2.166666667$$

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.

$$2x+2y=6,-5x+7y=11$$

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

$$2x+2y=6$$

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

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

Whakawehea ngā taha e rua ki te 2.

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

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

$$x=-y+3$$

Whakakapia te -y+3 mō te x ki tērā atu whārite, -5x+7y=11.

$$-5\left(-y+3\right)+7y=11$$

Whakareatia -5 ki te -y+3.

$$5y-15+7y=11$$

Tāpiri 5y ki te 7y.

$$12y-15=11$$

Me tāpiri 15 ki ngā taha e rua o te whārite.

$$12y=26$$

Whakawehea ngā taha e rua ki te 12.

$$y=\frac{13}{6}$$

Whakaurua te \frac{13}{6} mō y ki x=-y+3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{13}{6}+3$$

Tāpiri 3 ki te -\frac{13}{6}.

$$x=\frac{5}{6}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{5}{6},y=\frac{13}{6}$$

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.

$$2x+2y=6,-5x+7y=11$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&2\\-5&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}6\\11\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&2\\-5&7\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&2\\-5&7\end{matrix}\right))\left(\begin{matrix}2&2\\-5&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&2\\-5&7\end{matrix}\right))\left(\begin{matrix}6\\11\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&2\\-5&7\end{matrix}\right))\left(\begin{matrix}6\\11\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&2\\-5&7\end{matrix}\right))\left(\begin{matrix}6\\11\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{2\times 7-2\left(-5\right)}&-\frac{2}{2\times 7-2\left(-5\right)}\\-\frac{-5}{2\times 7-2\left(-5\right)}&\frac{2}{2\times 7-2\left(-5\right)}\end{matrix}\right)\left(\begin{matrix}6\\11\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{24}&-\frac{1}{12}\\\frac{5}{24}&\frac{1}{12}\end{matrix}\right)\left(\begin{matrix}6\\11\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{24}\times 6-\frac{1}{12}\times 11\\\frac{5}{24}\times 6+\frac{1}{12}\times 11\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{6}\\\frac{13}{6}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{5}{6},y=\frac{13}{6}$$

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.

$$2x+2y=6,-5x+7y=11$$

Kia ōrite ai a 2x me -5x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -5 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

$$-5\times 2x-5\times 2y=-5\times 6,2\left(-5\right)x+2\times 7y=2\times 11$$

Whakarūnātia.

$$-10x-10y=-30,-10x+14y=22$$

Me tango -10x+14y=22 mai i -10x-10y=-30 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-10x+10x-10y-14y=-30-22$$

Tāpiri -10x ki te 10x. Ka whakakore atu ngā kupu -10x me 10x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-10y-14y=-30-22$$

Tāpiri -10y ki te -14y.

$$-24y=-30-22$$

Tāpiri -30 ki te -22.

$$-24y=-52$$

Whakawehea ngā taha e rua ki te -24.

$$y=\frac{13}{6}$$

Whakaurua te \frac{13}{6} mō y ki -5x+7y=11. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-5x+7\times \frac{13}{6}=11$$

Whakareatia 7 ki te \frac{13}{6}.

$$-5x+\frac{91}{6}=11$$

Me tango \frac{91}{6} mai i ngā taha e rua o te whārite.

$$-5x=-\frac{25}{6}$$

Whakawehea ngā taha e rua ki te -5.

$$x=\frac{5}{6}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{5}{6},y=\frac{13}{6}$$