Whakaoti i te {l}{2x+4y=8}{3y-2x=6}

$$\left. \begin{array} { l } { 2 x + 4 y = 8 } \\ { 3 y - 2 x = 6 } \end{array} \right.$$

Whakaoti mō x, y

$$x=0$$

$$y=2$$

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.

$$2x+4y=8,-2x+3y=6$$

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

$$2x+4y=8$$

Me tango 4y mai i ngā taha e rua o te whārite.

$$2x=-4y+8$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(-4y+8\right)$$

Whakareatia \frac{1}{2} ki te -4y+8.

$$x=-2y+4$$

Whakakapia te -2y+4 mō te x ki tērā atu whārite, -2x+3y=6.

$$-2\left(-2y+4\right)+3y=6$$

Whakareatia -2 ki te -2y+4.

$$4y-8+3y=6$$

Tāpiri 4y ki te 3y.

$$7y-8=6$$

Me tāpiri 8 ki ngā taha e rua o te whārite.

$$7y=14$$

Whakawehea ngā taha e rua ki te 7.

$$y=2$$

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

$$x=-2\times 2+4$$

Whakareatia -2 ki te 2.

$$x=-4+4$$

Tāpiri 4 ki te -4.

$$x=0$$

Kua oti te pūnaha te whakatau.

$$x=0,y=2$$

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.

$$2x+4y=8,-2x+3y=6$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&4\\-2&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}8\\6\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}2&4\\-2&3\end{matrix}\right))\left(\begin{matrix}2&4\\-2&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&4\\-2&3\end{matrix}\right))\left(\begin{matrix}8\\6\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&4\\-2&3\end{matrix}\right))\left(\begin{matrix}8\\6\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&4\\-2&3\end{matrix}\right))\left(\begin{matrix}8\\6\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{2\times 3-4\left(-2\right)}&-\frac{4}{2\times 3-4\left(-2\right)}\\-\frac{-2}{2\times 3-4\left(-2\right)}&\frac{2}{2\times 3-4\left(-2\right)}\end{matrix}\right)\left(\begin{matrix}8\\6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{14}&-\frac{2}{7}\\\frac{1}{7}&\frac{1}{7}\end{matrix}\right)\left(\begin{matrix}8\\6\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{14}\times 8-\frac{2}{7}\times 6\\\frac{1}{7}\times 8+\frac{1}{7}\times 6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}0\\2\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=0,y=2$$

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.

$$2x+4y=8,-2x+3y=6$$

Kia ōrite ai a 2x 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 2.

$$-2\times 2x-2\times 4y=-2\times 8,2\left(-2\right)x+2\times 3y=2\times 6$$

Whakarūnātia.

$$-4x-8y=-16,-4x+6y=12$$

Me tango -4x+6y=12 mai i -4x-8y=-16 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-4x+4x-8y-6y=-16-12$$

Tāpiri -4x ki te 4x. Ka whakakore atu ngā kupu -4x me 4x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-8y-6y=-16-12$$

Tāpiri -8y ki te -6y.

$$-14y=-16-12$$

Tāpiri -16 ki te -12.

$$-14y=-28$$

Whakawehea ngā taha e rua ki te -14.

$$y=2$$

Whakaurua te 2 mō y ki -2x+3y=6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-2x+3\times 2=6$$

Whakareatia 3 ki te 2.

$$-2x+6=6$$

Me tango 6 mai i ngā taha e rua o te whārite.

$$-2x=0$$

Whakawehea ngā taha e rua ki te -2.

$$x=0$$

Kua oti te pūnaha te whakatau.

$$x=0,y=2$$