Whakaoti i te {l}{5x+4y=-3}{6x+3y=-2}

$$\left\{ \begin{array} { l } { 5 x + 4 y = - 3 } \\ { 6 x + 3 y = - 2 } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{1}{9}\approx 0.111111111$$

$$y=-\frac{8}{9}\approx -0.888888889$$

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.

$$5x+4y=-3,6x+3y=-2$$

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

$$5x+4y=-3$$

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

$$5x=-4y-3$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\left(-4y-3\right)$$

Whakareatia \frac{1}{5} ki te -4y-3.

$$x=-\frac{4}{5}y-\frac{3}{5}$$

Whakakapia te \frac{-4y-3}{5} mō te x ki tērā atu whārite, 6x+3y=-2.

$$6\left(-\frac{4}{5}y-\frac{3}{5}\right)+3y=-2$$

Whakareatia 6 ki te \frac{-4y-3}{5}.

$$-\frac{24}{5}y-\frac{18}{5}+3y=-2$$

Tāpiri -\frac{24y}{5} ki te 3y.

$$-\frac{9}{5}y-\frac{18}{5}=-2$$

Me tāpiri \frac{18}{5} ki ngā taha e rua o te whārite.

$$-\frac{9}{5}y=\frac{8}{5}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{9}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-\frac{8}{9}$$

Whakaurua te -\frac{8}{9} mō y ki x=-\frac{4}{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.

$$x=-\frac{4}{5}\left(-\frac{8}{9}\right)-\frac{3}{5}$$

Whakareatia -\frac{4}{5} ki te -\frac{8}{9} 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.

$$x=\frac{32}{45}-\frac{3}{5}$$

Tāpiri -\frac{3}{5} ki te \frac{32}{45} 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.

$$x=\frac{1}{9}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{1}{9},y=-\frac{8}{9}$$

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.

$$5x+4y=-3,6x+3y=-2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&4\\6&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-3\\-2\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&4\\6&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&4\\6&3\end{matrix}\right))\left(\begin{matrix}5&4\\6&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&4\\6&3\end{matrix}\right))\left(\begin{matrix}-3\\-2\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&4\\6&3\end{matrix}\right))\left(\begin{matrix}-3\\-2\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&4\\6&3\end{matrix}\right))\left(\begin{matrix}-3\\-2\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{5\times 3-4\times 6}&-\frac{4}{5\times 3-4\times 6}\\-\frac{6}{5\times 3-4\times 6}&\frac{5}{5\times 3-4\times 6}\end{matrix}\right)\left(\begin{matrix}-3\\-2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{3}&\frac{4}{9}\\\frac{2}{3}&-\frac{5}{9}\end{matrix}\right)\left(\begin{matrix}-3\\-2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{3}\left(-3\right)+\frac{4}{9}\left(-2\right)\\\frac{2}{3}\left(-3\right)-\frac{5}{9}\left(-2\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{9}\\-\frac{8}{9}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{1}{9},y=-\frac{8}{9}$$

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.

$$5x+4y=-3,6x+3y=-2$$

Kia ōrite ai a 5x me 6x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 6 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 5.

$$6\times 5x+6\times 4y=6\left(-3\right),5\times 6x+5\times 3y=5\left(-2\right)$$

Whakarūnātia.

$$30x+24y=-18,30x+15y=-10$$

Me tango 30x+15y=-10 mai i 30x+24y=-18 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$30x-30x+24y-15y=-18+10$$

Tāpiri 30x ki te -30x. Ka whakakore atu ngā kupu 30x me -30x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$24y-15y=-18+10$$

Tāpiri 24y ki te -15y.

$$9y=-18+10$$

Tāpiri -18 ki te 10.

$$9y=-8$$

Whakawehea ngā taha e rua ki te 9.

$$y=-\frac{8}{9}$$

Whakaurua te -\frac{8}{9} mō y ki 6x+3y=-2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$6x+3\left(-\frac{8}{9}\right)=-2$$

Whakareatia 3 ki te -\frac{8}{9}.

$$6x-\frac{8}{3}=-2$$

Me tāpiri \frac{8}{3} ki ngā taha e rua o te whārite.

$$6x=\frac{2}{3}$$

Whakawehea ngā taha e rua ki te 6.

$$x=\frac{1}{9}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{1}{9},y=-\frac{8}{9}$$