Whakaoti i te {l}{5x-6y=10}{2x+7y=3}

$$\left. \begin{array} { l } { 5 x - 6 y = 10 } \\ { 2 x + 7 y = 3 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{88}{47} = 1\frac{41}{47} \approx 1.872340426$$

$$y=-\frac{5}{47}\approx -0.106382979$$

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.

$$5x-6y=10,2x+7y=3$$

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

$$5x-6y=10$$

Me tāpiri 6y ki ngā taha e rua o te whārite.

$$5x=6y+10$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\left(6y+10\right)$$

Whakareatia \frac{1}{5} ki te 6y+10.

$$x=\frac{6}{5}y+2$$

Whakakapia te \frac{6y}{5}+2 mō te x ki tērā atu whārite, 2x+7y=3.

$$2\left(\frac{6}{5}y+2\right)+7y=3$$

Whakareatia 2 ki te \frac{6y}{5}+2.

$$\frac{12}{5}y+4+7y=3$$

Tāpiri \frac{12y}{5} ki te 7y.

$$\frac{47}{5}y+4=3$$

Me tango 4 mai i ngā taha e rua o te whārite.

$$\frac{47}{5}y=-1$$

Whakawehea ngā taha e rua o te whārite ki te \frac{47}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-\frac{5}{47}$$

Whakaurua te -\frac{5}{47} mō y ki x=\frac{6}{5}y+2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{6}{5}\left(-\frac{5}{47}\right)+2$$

Whakareatia \frac{6}{5} ki te -\frac{5}{47} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=-\frac{6}{47}+2$$

Tāpiri 2 ki te -\frac{6}{47}.

$$x=\frac{88}{47}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{88}{47},y=-\frac{5}{47}$$

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.

$$5x-6y=10,2x+7y=3$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&-6\\2&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}10\\3\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&-6\\2&7\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&-6\\2&7\end{matrix}\right))\left(\begin{matrix}5&-6\\2&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-6\\2&7\end{matrix}\right))\left(\begin{matrix}10\\3\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-6\\2&7\end{matrix}\right))\left(\begin{matrix}10\\3\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-6\\2&7\end{matrix}\right))\left(\begin{matrix}10\\3\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{5\times 7-\left(-6\times 2\right)}&-\frac{-6}{5\times 7-\left(-6\times 2\right)}\\-\frac{2}{5\times 7-\left(-6\times 2\right)}&\frac{5}{5\times 7-\left(-6\times 2\right)}\end{matrix}\right)\left(\begin{matrix}10\\3\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{47}&\frac{6}{47}\\-\frac{2}{47}&\frac{5}{47}\end{matrix}\right)\left(\begin{matrix}10\\3\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{47}\times 10+\frac{6}{47}\times 3\\-\frac{2}{47}\times 10+\frac{5}{47}\times 3\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{88}{47}\\-\frac{5}{47}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{88}{47},y=-\frac{5}{47}$$

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.

$$5x-6y=10,2x+7y=3$$

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

$$2\times 5x+2\left(-6\right)y=2\times 10,5\times 2x+5\times 7y=5\times 3$$

Whakarūnātia.

$$10x-12y=20,10x+35y=15$$

Me tango 10x+35y=15 mai i 10x-12y=20 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$10x-10x-12y-35y=20-15$$

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.

$$-12y-35y=20-15$$

Tāpiri -12y ki te -35y.

$$-47y=20-15$$

Tāpiri 20 ki te -15.

$$-47y=5$$

Whakawehea ngā taha e rua ki te -47.

$$y=-\frac{5}{47}$$

Whakaurua te -\frac{5}{47} mō y ki 2x+7y=3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x+7\left(-\frac{5}{47}\right)=3$$

Whakareatia 7 ki te -\frac{5}{47}.

$$2x-\frac{35}{47}=3$$

Me tāpiri \frac{35}{47} ki ngā taha e rua o te whārite.

$$2x=\frac{176}{47}$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{88}{47}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{88}{47},y=-\frac{5}{47}$$