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

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

Whakaoti mō x, y

$$x=\frac{1}{2}=0.5$$

$$y=\frac{3}{4}=0.75$$

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.

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

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

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

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

$$2x=4y-2$$

Whakawehea ngā taha e rua ki te 2.

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

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

$$x=2y-1$$

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

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

Whakareatia 3 ki te 2y-1.

$$6y-3+2y=3$$

Tāpiri 6y ki te 2y.

$$8y-3=3$$

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

$$8y=6$$

Whakawehea ngā taha e rua ki te 8.

$$y=\frac{3}{4}$$

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

$$x=2\times \frac{3}{4}-1$$

Whakareatia 2 ki te \frac{3}{4}.

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

Tāpiri -1 ki te \frac{3}{2}.

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

Kua oti te pūnaha te whakatau.

$$x=\frac{1}{2},y=\frac{3}{4}$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=\frac{1}{2},y=\frac{3}{4}$$

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.

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

Kia ōrite ai a 2x 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 2.

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

Whakarūnātia.

$$6x-12y=-6,6x+4y=6$$

Me tango 6x+4y=6 mai i 6x-12y=-6 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x-12y-4y=-6-6$$

Tāpiri 6x ki te -6x. Ka whakakore atu ngā kupu 6x me -6x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-12y-4y=-6-6$$

Tāpiri -12y ki te -4y.

$$-16y=-6-6$$

Tāpiri -6 ki te -6.

$$-16y=-12$$

Whakawehea ngā taha e rua ki te -16.

$$y=\frac{3}{4}$$

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

$$3x+2\times \frac{3}{4}=3$$

Whakareatia 2 ki te \frac{3}{4}.

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

Me tango \frac{3}{2} mai i ngā taha e rua o te whārite.

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

Whakawehea ngā taha e rua ki te 3.

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

Kua oti te pūnaha te whakatau.

$$x=\frac{1}{2},y=\frac{3}{4}$$