Whakaoti i te {l}{5x+6y=32}{3x-2y=-20}

$$\left\{ \begin{array} { l } { 5 x + 6 y = 32 } \\ { 3 x - 2 y = - 20 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-2$$

$$y=7$$

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.

$$5x+6y=32,3x-2y=-20$$

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

$$5x+6y=32$$

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

$$5x=-6y+32$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\left(-6y+32\right)$$

Whakareatia \frac{1}{5} ki te -6y+32.

$$x=-\frac{6}{5}y+\frac{32}{5}$$

Whakakapia te \frac{-6y+32}{5} mō te x ki tērā atu whārite, 3x-2y=-20.

$$3\left(-\frac{6}{5}y+\frac{32}{5}\right)-2y=-20$$

Whakareatia 3 ki te \frac{-6y+32}{5}.

$$-\frac{18}{5}y+\frac{96}{5}-2y=-20$$

Tāpiri -\frac{18y}{5} ki te -2y.

$$-\frac{28}{5}y+\frac{96}{5}=-20$$

Me tango \frac{96}{5} mai i ngā taha e rua o te whārite.

$$-\frac{28}{5}y=-\frac{196}{5}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{28}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=7$$

Whakaurua te 7 mō y ki x=-\frac{6}{5}y+\frac{32}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{6}{5}\times 7+\frac{32}{5}$$

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

$$x=\frac{-42+32}{5}$$

Tāpiri \frac{32}{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.

$$x=-2$$

Kua oti te pūnaha te whakatau.

$$x=-2,y=7$$

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.

$$5x+6y=32,3x-2y=-20$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&6\\3&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}32\\-20\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&6\\3&-2\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&6\\3&-2\end{matrix}\right))\left(\begin{matrix}5&6\\3&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&6\\3&-2\end{matrix}\right))\left(\begin{matrix}32\\-20\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&6\\3&-2\end{matrix}\right))\left(\begin{matrix}32\\-20\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&6\\3&-2\end{matrix}\right))\left(\begin{matrix}32\\-20\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{5\left(-2\right)-6\times 3}&-\frac{6}{5\left(-2\right)-6\times 3}\\-\frac{3}{5\left(-2\right)-6\times 3}&\frac{5}{5\left(-2\right)-6\times 3}\end{matrix}\right)\left(\begin{matrix}32\\-20\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{14}&\frac{3}{14}\\\frac{3}{28}&-\frac{5}{28}\end{matrix}\right)\left(\begin{matrix}32\\-20\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{14}\times 32+\frac{3}{14}\left(-20\right)\\\frac{3}{28}\times 32-\frac{5}{28}\left(-20\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=-2,y=7$$

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.

$$5x+6y=32,3x-2y=-20$$

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.

$$3\times 5x+3\times 6y=3\times 32,5\times 3x+5\left(-2\right)y=5\left(-20\right)$$

Whakarūnātia.

$$15x+18y=96,15x-10y=-100$$

Me tango 15x-10y=-100 mai i 15x+18y=96 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$15x-15x+18y+10y=96+100$$

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.

$$18y+10y=96+100$$

Tāpiri 18y ki te 10y.

$$28y=96+100$$

Tāpiri 96 ki te 100.

$$28y=196$$

Whakawehea ngā taha e rua ki te 28.

$$y=7$$

Whakaurua te 7 mō y ki 3x-2y=-20. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x-2\times 7=-20$$

Whakareatia -2 ki te 7.

$$3x-14=-20$$

Me tāpiri 14 ki ngā taha e rua o te whārite.

$$3x=-6$$

Whakawehea ngā taha e rua ki te 3.

$$x=-2$$

Kua oti te pūnaha te whakatau.

$$x=-2,y=7$$