Whakaoti i te {l}{5x-7y=-27}{2x+3y=24}

$$\left. \begin{array} { l } { 5 x - 7 y = - 27 } \\ { 2 x + 3 y = 24 } \end{array} \right.$$

Whakaoti mō x, y

$$x=3$$

$$y=6$$

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.

$$5x-7y=-27,2x+3y=24$$

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

$$5x-7y=-27$$

Me tāpiri 7y ki ngā taha e rua o te whārite.

$$5x=7y-27$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\left(7y-27\right)$$

Whakareatia \frac{1}{5} ki te 7y-27.

$$x=\frac{7}{5}y-\frac{27}{5}$$

Whakakapia te \frac{7y-27}{5} mō te x ki tērā atu whārite, 2x+3y=24.

$$2\left(\frac{7}{5}y-\frac{27}{5}\right)+3y=24$$

Whakareatia 2 ki te \frac{7y-27}{5}.

$$\frac{14}{5}y-\frac{54}{5}+3y=24$$

Tāpiri \frac{14y}{5} ki te 3y.

$$\frac{29}{5}y-\frac{54}{5}=24$$

Me tāpiri \frac{54}{5} ki ngā taha e rua o te whārite.

$$\frac{29}{5}y=\frac{174}{5}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{29}{5}, 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{7}{5}y-\frac{27}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{7}{5}\times 6-\frac{27}{5}$$

Whakareatia \frac{7}{5} ki te 6.

$$x=\frac{42-27}{5}$$

Tāpiri -\frac{27}{5} ki te \frac{42}{5} 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.

$$x=3,y=6$$

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.

$$5x-7y=-27,2x+3y=24$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&-7\\2&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-27\\24\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&-7\\2&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&-7\\2&3\end{matrix}\right))\left(\begin{matrix}5&-7\\2&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-7\\2&3\end{matrix}\right))\left(\begin{matrix}-27\\24\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-7\\2&3\end{matrix}\right))\left(\begin{matrix}-27\\24\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-7\\2&3\end{matrix}\right))\left(\begin{matrix}-27\\24\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{5\times 3-\left(-7\times 2\right)}&-\frac{-7}{5\times 3-\left(-7\times 2\right)}\\-\frac{2}{5\times 3-\left(-7\times 2\right)}&\frac{5}{5\times 3-\left(-7\times 2\right)}\end{matrix}\right)\left(\begin{matrix}-27\\24\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{29}&\frac{7}{29}\\-\frac{2}{29}&\frac{5}{29}\end{matrix}\right)\left(\begin{matrix}-27\\24\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{29}\left(-27\right)+\frac{7}{29}\times 24\\-\frac{2}{29}\left(-27\right)+\frac{5}{29}\times 24\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3\\6\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=3,y=6$$

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.

$$5x-7y=-27,2x+3y=24$$

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.

$$2\times 5x+2\left(-7\right)y=2\left(-27\right),5\times 2x+5\times 3y=5\times 24$$

Whakarūnātia.

$$10x-14y=-54,10x+15y=120$$

Me tango 10x+15y=120 mai i 10x-14y=-54 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$10x-10x-14y-15y=-54-120$$

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.

$$-14y-15y=-54-120$$

Tāpiri -14y ki te -15y.

$$-29y=-54-120$$

Tāpiri -54 ki te -120.

$$-29y=-174$$

Whakawehea ngā taha e rua ki te -29.

$$y=6$$

Whakaurua te 6 mō y ki 2x+3y=24. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x+3\times 6=24$$

Whakareatia 3 ki te 6.

$$2x+18=24$$

Me tango 18 mai i ngā taha e rua o te whārite.

$$2x=6$$

Whakawehea ngā taha e rua ki te 2.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=6$$