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

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

Whakaoti mō x, y

$$x=4$$

$$y=-4$$

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.

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

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

$$2x+y=4$$

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

$$2x=-y+4$$

Whakawehea ngā taha e rua ki te 2.

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

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

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

Whakakapia te -\frac{y}{2}+2 mō te x ki tērā atu whārite, 3x+2y=4.

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

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

$$-\frac{3}{2}y+6+2y=4$$

Tāpiri -\frac{3y}{2} ki te 2y.

$$\frac{1}{2}y+6=4$$

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

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

Me whakarea ngā taha e rua ki te 2.

$$y=-4$$

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

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

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

$$x=2+2$$

Tāpiri 2 ki te 2.

$$x=4$$

Kua oti te pūnaha te whakatau.

$$x=4,y=-4$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=4,y=-4$$

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.

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

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.

$$3\times 2x+3y=3\times 4,2\times 3x+2\times 2y=2\times 4$$

Whakarūnātia.

$$6x+3y=12,6x+4y=8$$

Me tango 6x+4y=8 mai i 6x+3y=12 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x+3y-4y=12-8$$

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.

$$3y-4y=12-8$$

Tāpiri 3y ki te -4y.

$$-y=12-8$$

Tāpiri 12 ki te -8.

$$-y=4$$

Whakawehea ngā taha e rua ki te -1.

$$y=-4$$

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

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

Whakareatia 2 ki te -4.

$$3x-8=4$$

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

$$3x=12$$

Whakawehea ngā taha e rua ki te 3.

$$x=4$$

Kua oti te pūnaha te whakatau.

$$x=4,y=-4$$