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

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

Whakaoti mō x, y

$$x=-3$$

$$y=2$$

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.

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

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

$$5x+6y=-3$$

Me tango 6y mai i ngā taha e rua o te whārite.

$$5x=-6y-3$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\left(-6y-3\right)$$

Whakareatia \frac{1}{5} ki te -6y-3.

$$x=-\frac{6}{5}y-\frac{3}{5}$$

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

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

Whakareatia 3 ki te \frac{-6y-3}{5}.

$$-\frac{18}{5}y-\frac{9}{5}+7y=5$$

Tāpiri -\frac{18y}{5} ki te 7y.

$$\frac{17}{5}y-\frac{9}{5}=5$$

Me tāpiri \frac{9}{5} ki ngā taha e rua o te whārite.

$$\frac{17}{5}y=\frac{34}{5}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{17}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=2$$

Whakaurua te 2 mō y ki x=-\frac{6}{5}y-\frac{3}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{6}{5}\times 2-\frac{3}{5}$$

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

$$x=\frac{-12-3}{5}$$

Tāpiri -\frac{3}{5} ki te -\frac{12}{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.

$$x=-3,y=2$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&6\\3&7\end{matrix}\right).

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{17}&-\frac{6}{17}\\-\frac{3}{17}&\frac{5}{17}\end{matrix}\right)\left(\begin{matrix}-3\\5\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{17}\left(-3\right)-\frac{6}{17}\times 5\\-\frac{3}{17}\left(-3\right)+\frac{5}{17}\times 5\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-3\\2\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-3,y=2$$

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.

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

Kia ōrite ai a 5x me 3x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 3 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 5.

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

Whakarūnātia.

$$15x+18y=-9,15x+35y=25$$

Me tango 15x+35y=25 mai i 15x+18y=-9 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$15x-15x+18y-35y=-9-25$$

Tāpiri 15x ki te -15x. Ka whakakore atu ngā kupu 15x me -15x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$18y-35y=-9-25$$

Tāpiri 18y ki te -35y.

$$-17y=-9-25$$

Tāpiri -9 ki te -25.

$$-17y=-34$$

Whakawehea ngā taha e rua ki te -17.

$$y=2$$

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

$$3x+7\times 2=5$$

Whakareatia 7 ki te 2.

$$3x+14=5$$

Me tango 14 mai i ngā taha e rua o te whārite.

$$3x=-9$$

Whakawehea ngā taha e rua ki te 3.

$$x=-3$$

Kua oti te pūnaha te whakatau.

$$x=-3,y=2$$