Whakaoti i te {l}{4x+2y=-2}{2x+3y=-7}

$$\left\{ \begin{array} { l } { 4 x + 2 y = - 2 } \\ { 2 x + 3 y = - 7 } \end{array} \right.$$

Whakaoti mō x, y

$$x=1$$

$$y=-3$$

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.

$$4x+2y=-2,2x+3y=-7$$

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

$$4x+2y=-2$$

Me tango 2y mai i ngā taha e rua o te whārite.

$$4x=-2y-2$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{1}{4}\left(-2y-2\right)$$

Whakareatia \frac{1}{4} ki te -2y-2.

$$x=-\frac{1}{2}y-\frac{1}{2}$$

Whakakapia te \frac{-y-1}{2} mō te x ki tērā atu whārite, 2x+3y=-7.

$$2\left(-\frac{1}{2}y-\frac{1}{2}\right)+3y=-7$$

Whakareatia 2 ki te \frac{-y-1}{2}.

$$-y-1+3y=-7$$

Tāpiri -y ki te 3y.

$$2y-1=-7$$

Me tāpiri 1 ki ngā taha e rua o te whārite.

$$2y=-6$$

Whakawehea ngā taha e rua ki te 2.

$$y=-3$$

Whakaurua te -3 mō y ki x=-\frac{1}{2}y-\frac{1}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{2}\left(-3\right)-\frac{1}{2}$$

Whakareatia -\frac{1}{2} ki te -3.

$$x=\frac{3-1}{2}$$

Tāpiri -\frac{1}{2} ki te \frac{3}{2} 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.

$$x=1,y=-3$$

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.

$$4x+2y=-2,2x+3y=-7$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4&2\\2&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&2\\2&3\end{matrix}\right))\left(\begin{matrix}4&2\\2&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&2\\2&3\end{matrix}\right))\left(\begin{matrix}-2\\-7\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&2\\2&3\end{matrix}\right))\left(\begin{matrix}-2\\-7\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&2\\2&3\end{matrix}\right))\left(\begin{matrix}-2\\-7\end{matrix}\right)$$

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1\\-3\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=1,y=-3$$

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.

$$4x+2y=-2,2x+3y=-7$$

Kia ōrite ai a 4x 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 4.

$$2\times 4x+2\times 2y=2\left(-2\right),4\times 2x+4\times 3y=4\left(-7\right)$$

Whakarūnātia.

$$8x+4y=-4,8x+12y=-28$$

Me tango 8x+12y=-28 mai i 8x+4y=-4 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$8x-8x+4y-12y=-4+28$$

Tāpiri 8x ki te -8x. Ka whakakore atu ngā kupu 8x me -8x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$4y-12y=-4+28$$

Tāpiri 4y ki te -12y.

$$-8y=-4+28$$

Tāpiri -4 ki te 28.

$$-8y=24$$

Whakawehea ngā taha e rua ki te -8.

$$y=-3$$

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

$$2x+3\left(-3\right)=-7$$

Whakareatia 3 ki te -3.

$$2x-9=-7$$

Me tāpiri 9 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.

$$x=1,y=-3$$