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

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

Whakaoti mō x, y

$$x=-1$$

$$y=1$$

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.

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

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 tango 3y mai i 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, 2x-2y=-4.

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

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

$$-\frac{6}{5}y-\frac{4}{5}-2y=-4$$

Tāpiri -\frac{6y}{5} ki te -2y.

$$-\frac{16}{5}y-\frac{4}{5}=-4$$

Me tāpiri \frac{4}{5} ki ngā taha e rua o te whārite.

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

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.

$$y=1$$

Whakaurua te 1 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.

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

Tāpiri -\frac{2}{5} ki te -\frac{3}{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.

$$x=-1$$

Kua oti te pūnaha te whakatau.

$$x=-1,y=1$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=-1,y=1$$

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.

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

Kia ōrite ai a 5x me 2x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 5.

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

Whakarūnātia.

$$10x+6y=-4,10x-10y=-20$$

Me tango 10x-10y=-20 mai i 10x+6y=-4 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$10x-10x+6y+10y=-4+20$$

Tāpiri 10x ki te -10x. Ka whakakore atu ngā kupu 10x me -10x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$6y+10y=-4+20$$

Tāpiri 6y ki te 10y.

$$16y=-4+20$$

Tāpiri -4 ki te 20.

$$16y=16$$

Whakawehea ngā taha e rua ki te 16.

$$y=1$$

Whakaurua te 1 mō y ki 2x-2y=-4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x-2=-4$$

Me tāpiri 2 ki ngā taha e rua o te whārite.

$$2x=-2$$

Whakawehea ngā taha e rua ki te 2.

$$x=-1$$

Kua oti te pūnaha te whakatau.

$$x=-1,y=1$$