Whakaoti i te {l}{x+2y=-2}{4y=1-3x}

$$\left\{ \begin{array} { l } { x + 2 y = - 2 } \\ { 4 y = 1 - 3 x } \end{array} \right.$$

Whakaoti mō x, y

$$x=5$$

$$y = -\frac{7}{2} = -3\frac{1}{2} = -3.5$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Me tāpiri te 3x ki ngā taha e rua.

$$4y+3x=1$$

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.

$$x+2y=-2,3x+4y=1$$

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

$$x+2y=-2$$

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

$$x=-2y-2$$

Whakakapia te -2y-2 mō te x ki tērā atu whārite, 3x+4y=1.

$$3\left(-2y-2\right)+4y=1$$

Whakareatia 3 ki te -2y-2.

$$-6y-6+4y=1$$

Tāpiri -6y ki te 4y.

$$-2y-6=1$$

Me tāpiri 6 ki ngā taha e rua o te whārite.

$$-2y=7$$

Whakawehea ngā taha e rua ki te -2.

$$y=-\frac{7}{2}$$

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

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

Whakareatia -2 ki te -\frac{7}{2}.

$$x=7-2$$

Tāpiri -2 ki te 7.

$$x=5$$

Kua oti te pūnaha te whakatau.

$$x=5,y=-\frac{7}{2}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Me tāpiri te 3x ki ngā taha e rua.

$$4y+3x=1$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$x+2y=-2,3x+4y=1$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&2\\3&4\end{matrix}\right).

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\\-\frac{7}{2}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=5,y=-\frac{7}{2}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Me tāpiri te 3x ki ngā taha e rua.

$$4y+3x=1$$

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.

$$x+2y=-2,3x+4y=1$$

Kia ōrite ai a x me 3x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 3 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$3x+3\times 2y=3\left(-2\right),3x+4y=1$$

Whakarūnātia.

$$3x+6y=-6,3x+4y=1$$

Me tango 3x+4y=1 mai i 3x+6y=-6 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3x-3x+6y-4y=-6-1$$

Tāpiri 3x ki te -3x. Ka whakakore atu ngā kupu 3x me -3x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$6y-4y=-6-1$$

Tāpiri 6y ki te -4y.

$$2y=-6-1$$

Tāpiri -6 ki te -1.

$$2y=-7$$

Whakawehea ngā taha e rua ki te 2.

$$y=-\frac{7}{2}$$

Whakaurua te -\frac{7}{2} mō y ki 3x+4y=1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x+4\left(-\frac{7}{2}\right)=1$$

Whakareatia 4 ki te -\frac{7}{2}.

$$3x-14=1$$

Me tāpiri 14 ki ngā taha e rua o te whārite.

$$3x=15$$

Whakawehea ngā taha e rua ki te 3.

$$x=5$$

Kua oti te pūnaha te whakatau.

$$x=5,y=-\frac{7}{2}$$