Whakaoti i te {l}{5x-4y=-7}{-6x+8y=2}

$$\left. \begin{array}{l}{ 5 x - 4 y = - 7 }\\{ - 6 x + 8 y = 2 }\end{array} \right.$$

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.

$$5x-4y=-7,-6x+8y=2$$

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

$$5x-4y=-7$$

Me tāpiri 4y ki ngā taha e rua o te whārite.

$$5x=4y-7$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\left(4y-7\right)$$

Whakareatia \frac{1}{5} ki te 4y-7.

$$x=\frac{4}{5}y-\frac{7}{5}$$

Whakakapia te \frac{4y-7}{5} mō te x ki tērā atu whārite, -6x+8y=2.

$$-6\left(\frac{4}{5}y-\frac{7}{5}\right)+8y=2$$

Whakareatia -6 ki te \frac{4y-7}{5}.

$$-\frac{24}{5}y+\frac{42}{5}+8y=2$$

Tāpiri -\frac{24y}{5} ki te 8y.

$$\frac{16}{5}y+\frac{42}{5}=2$$

Me tango \frac{42}{5} mai i ngā taha e rua o te whārite.

$$\frac{16}{5}y=-\frac{32}{5}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{16}{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{4}{5}y-\frac{7}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{4}{5}\left(-2\right)-\frac{7}{5}$$

Whakareatia \frac{4}{5} ki te -2.

$$x=\frac{-8-7}{5}$$

Tāpiri -\frac{7}{5} ki te -\frac{8}{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.

$$5x-4y=-7,-6x+8y=2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&-4\\-6&8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-7\\2\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&-4\\-6&8\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&-4\\-6&8\end{matrix}\right))\left(\begin{matrix}5&-4\\-6&8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-4\\-6&8\end{matrix}\right))\left(\begin{matrix}-7\\2\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-4\\-6&8\end{matrix}\right))\left(\begin{matrix}-7\\2\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-4\\-6&8\end{matrix}\right))\left(\begin{matrix}-7\\2\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{8}{5\times 8-\left(-4\left(-6\right)\right)}&-\frac{-4}{5\times 8-\left(-4\left(-6\right)\right)}\\-\frac{-6}{5\times 8-\left(-4\left(-6\right)\right)}&\frac{5}{5\times 8-\left(-4\left(-6\right)\right)}\end{matrix}\right)\left(\begin{matrix}-7\\2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}&\frac{1}{4}\\\frac{3}{8}&\frac{5}{16}\end{matrix}\right)\left(\begin{matrix}-7\\2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}\left(-7\right)+\frac{1}{4}\times 2\\\frac{3}{8}\left(-7\right)+\frac{5}{16}\times 2\end{matrix}\right)$$

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.

$$5x-4y=-7,-6x+8y=2$$

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.

$$-6\times 5x-6\left(-4\right)y=-6\left(-7\right),5\left(-6\right)x+5\times 8y=5\times 2$$

Whakarūnātia.

$$-30x+24y=42,-30x+40y=10$$

Me tango -30x+40y=10 mai i -30x+24y=42 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-30x+30x+24y-40y=42-10$$

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.

$$24y-40y=42-10$$

Tāpiri 24y ki te -40y.

$$-16y=42-10$$

Tāpiri 42 ki te -10.

$$-16y=32$$

Whakawehea ngā taha e rua ki te -16.

$$y=-2$$

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

$$-6x+8\left(-2\right)=2$$

Whakareatia 8 ki te -2.

$$-6x-16=2$$

Me tāpiri 16 ki ngā taha e rua o te whārite.

$$-6x=18$$

Whakawehea ngā taha e rua ki te -6.

$$x=-3$$

Kua oti te pūnaha te whakatau.

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