Whakaoti i te {l}{2x+3y=4}{3x+4y=10}

$$\left. \begin{array} { l } { 2 x + 3 y = 4 } \\ { 3 x + 4 y = 10 } \end{array} \right.$$

Whakaoti mō x, y

$$x=14$$

$$y=-8$$

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.

$$2x+3y=4,3x+4y=10$$

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

$$2x+3y=4$$

Me tango 3y mai i ngā taha e rua o te whārite.

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

Whakawehea ngā taha e rua ki te 2.

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

Whakareatia \frac{1}{2} ki te -3y+4.

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

Whakakapia te -\frac{3y}{2}+2 mō te x ki tērā atu whārite, 3x+4y=10.

$$3\left(-\frac{3}{2}y+2\right)+4y=10$$

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

$$-\frac{9}{2}y+6+4y=10$$

Tāpiri -\frac{9y}{2} ki te 4y.

$$-\frac{1}{2}y+6=10$$

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

$$-\frac{1}{2}y=4$$

Me whakarea ngā taha e rua ki te -2.

$$y=-8$$

Whakaurua te -8 mō y ki x=-\frac{3}{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.

$$x=-\frac{3}{2}\left(-8\right)+2$$

Whakareatia -\frac{3}{2} ki te -8.

$$x=12+2$$

Tāpiri 2 ki te 12.

$$x=14$$

Kua oti te pūnaha te whakatau.

$$x=14,y=-8$$

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.

$$2x+3y=4,3x+4y=10$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&3\\3&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\\10\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}2&3\\3&4\end{matrix}\right))\left(\begin{matrix}2&3\\3&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\3&4\end{matrix}\right))\left(\begin{matrix}4\\10\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\3&4\end{matrix}\right))\left(\begin{matrix}4\\10\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\3&4\end{matrix}\right))\left(\begin{matrix}4\\10\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{2\times 4-3\times 3}&-\frac{3}{2\times 4-3\times 3}\\-\frac{3}{2\times 4-3\times 3}&\frac{2}{2\times 4-3\times 3}\end{matrix}\right)\left(\begin{matrix}4\\10\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-4&3\\3&-2\end{matrix}\right)\left(\begin{matrix}4\\10\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-4\times 4+3\times 10\\3\times 4-2\times 10\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}14\\-8\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=14,y=-8$$

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.

$$2x+3y=4,3x+4y=10$$

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.

$$3\times 2x+3\times 3y=3\times 4,2\times 3x+2\times 4y=2\times 10$$

Whakarūnātia.

$$6x+9y=12,6x+8y=20$$

Me tango 6x+8y=20 mai i 6x+9y=12 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x+9y-8y=12-20$$

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.

$$9y-8y=12-20$$

Tāpiri 9y ki te -8y.

$$y=12-20$$

Tāpiri 12 ki te -20.

$$y=-8$$

Whakaurua te -8 mō y ki 3x+4y=10. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x+4\left(-8\right)=10$$

Whakareatia 4 ki te -8.

$$3x-32=10$$

Me tāpiri 32 ki ngā taha e rua o te whārite.

$$3x=42$$

Whakawehea ngā taha e rua ki te 3.

$$x=14$$

Kua oti te pūnaha te whakatau.

$$x=14,y=-8$$