Whakaoti i te {l}{6x+5y=27}{2x+y=13}

$$\left. \begin{array} { l } { 6 x + 5 y = 27 } \\ { 2 x + y = 13 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{19}{2} = 9\frac{1}{2} = 9.5$$

$$y=-6$$

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.

$$6x+5y=27,2x+y=13$$

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

$$6x+5y=27$$

Me tango 5y mai i ngā taha e rua o te whārite.

$$6x=-5y+27$$

Whakawehea ngā taha e rua ki te 6.

$$x=\frac{1}{6}\left(-5y+27\right)$$

Whakareatia \frac{1}{6} ki te -5y+27.

$$x=-\frac{5}{6}y+\frac{9}{2}$$

Whakakapia te -\frac{5y}{6}+\frac{9}{2} mō te x ki tērā atu whārite, 2x+y=13.

$$2\left(-\frac{5}{6}y+\frac{9}{2}\right)+y=13$$

Whakareatia 2 ki te -\frac{5y}{6}+\frac{9}{2}.

$$-\frac{5}{3}y+9+y=13$$

Tāpiri -\frac{5y}{3} ki te y.

$$-\frac{2}{3}y+9=13$$

Me tango 9 mai i ngā taha e rua o te whārite.

$$-\frac{2}{3}y=4$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{2}{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{5}{6}y+\frac{9}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{5}{6}\left(-6\right)+\frac{9}{2}$$

Whakareatia -\frac{5}{6} ki te -6.

$$x=5+\frac{9}{2}$$

Tāpiri \frac{9}{2} ki te 5.

$$x=\frac{19}{2}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{19}{2},y=-6$$

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.

$$6x+5y=27,2x+y=13$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}6&5\\2&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}27\\13\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}6&5\\2&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}6&5\\2&1\end{matrix}\right))\left(\begin{matrix}6&5\\2&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&5\\2&1\end{matrix}\right))\left(\begin{matrix}27\\13\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&5\\2&1\end{matrix}\right))\left(\begin{matrix}27\\13\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&5\\2&1\end{matrix}\right))\left(\begin{matrix}27\\13\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{6-5\times 2}&-\frac{5}{6-5\times 2}\\-\frac{2}{6-5\times 2}&\frac{6}{6-5\times 2}\end{matrix}\right)\left(\begin{matrix}27\\13\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{4}&\frac{5}{4}\\\frac{1}{2}&-\frac{3}{2}\end{matrix}\right)\left(\begin{matrix}27\\13\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{4}\times 27+\frac{5}{4}\times 13\\\frac{1}{2}\times 27-\frac{3}{2}\times 13\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{19}{2}\\-6\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{19}{2},y=-6$$

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.

$$6x+5y=27,2x+y=13$$

Kia ōrite ai a 6x 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 6.

$$2\times 6x+2\times 5y=2\times 27,6\times 2x+6y=6\times 13$$

Whakarūnātia.

$$12x+10y=54,12x+6y=78$$

Me tango 12x+6y=78 mai i 12x+10y=54 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$12x-12x+10y-6y=54-78$$

Tāpiri 12x ki te -12x. Ka whakakore atu ngā kupu 12x me -12x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$10y-6y=54-78$$

Tāpiri 10y ki te -6y.

$$4y=54-78$$

Tāpiri 54 ki te -78.

$$4y=-24$$

Whakawehea ngā taha e rua ki te 4.

$$y=-6$$

Whakaurua te -6 mō y ki 2x+y=13. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x-6=13$$

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

$$2x=19$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{19}{2}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{19}{2},y=-6$$