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

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

Whakaoti mō x, y

$$x=-\frac{11}{28}\approx -0.392857143$$

$$y = -\frac{37}{28} = -1\frac{9}{28} \approx -1.321428571$$

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.

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

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

$$5x-3y=2$$

Me tāpiri 3y ki ngā taha e rua o te whārite.

$$5x=3y+2$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\left(3y+2\right)$$

Whakareatia \frac{1}{5} ki te 3y+2.

$$x=\frac{3}{5}y+\frac{2}{5}$$

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

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

Whakareatia 6 ki te \frac{3y+2}{5}.

$$\frac{18}{5}y+\frac{12}{5}+2y=-5$$

Tāpiri \frac{18y}{5} ki te 2y.

$$\frac{28}{5}y+\frac{12}{5}=-5$$

Me tango \frac{12}{5} mai i ngā taha e rua o te whārite.

$$\frac{28}{5}y=-\frac{37}{5}$$

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.

$$y=-\frac{37}{28}$$

Whakaurua te -\frac{37}{28} mō y ki x=\frac{3}{5}y+\frac{2}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{3}{5}\left(-\frac{37}{28}\right)+\frac{2}{5}$$

Whakareatia \frac{3}{5} ki te -\frac{37}{28} 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.

$$x=-\frac{111}{140}+\frac{2}{5}$$

Tāpiri \frac{2}{5} ki te -\frac{111}{140} 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.

$$x=-\frac{11}{28}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{11}{28},y=-\frac{37}{28}$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{14}&\frac{3}{28}\\-\frac{3}{14}&\frac{5}{28}\end{matrix}\right)\left(\begin{matrix}2\\-5\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{14}\times 2+\frac{3}{28}\left(-5\right)\\-\frac{3}{14}\times 2+\frac{5}{28}\left(-5\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{11}{28}\\-\frac{37}{28}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{11}{28},y=-\frac{37}{28}$$

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.

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

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.

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

Whakarūnātia.

$$30x-18y=12,30x+10y=-25$$

Me tango 30x+10y=-25 mai i 30x-18y=12 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$30x-30x-18y-10y=12+25$$

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.

$$-18y-10y=12+25$$

Tāpiri -18y ki te -10y.

$$-28y=12+25$$

Tāpiri 12 ki te 25.

$$-28y=37$$

Whakawehea ngā taha e rua ki te -28.

$$y=-\frac{37}{28}$$

Whakaurua te -\frac{37}{28} mō y ki 6x+2y=-5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$6x+2\left(-\frac{37}{28}\right)=-5$$

Whakareatia 2 ki te -\frac{37}{28}.

$$6x-\frac{37}{14}=-5$$

Me tāpiri \frac{37}{14} ki ngā taha e rua o te whārite.

$$6x=-\frac{33}{14}$$

Whakawehea ngā taha e rua ki te 6.

$$x=-\frac{11}{28}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{11}{28},y=-\frac{37}{28}$$