Whakaoti i te {l}{3x-4y=-6}{2x+4y=16}

$$\left. \begin{array} { l } { 3 x - 4 y = - 6 } \\ { 2 x + 4 y = 16 } \end{array} \right.$$

Whakaoti mō x, y

$$x=2$$

$$y=3$$

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.

$$3x-4y=-6,2x+4y=16$$

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

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

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

$$3x=4y-6$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(4y-6\right)$$

Whakareatia \frac{1}{3} ki te 4y-6.

$$x=\frac{4}{3}y-2$$

Whakakapia te \frac{4y}{3}-2 mō te x ki tērā atu whārite, 2x+4y=16.

$$2\left(\frac{4}{3}y-2\right)+4y=16$$

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

$$\frac{8}{3}y-4+4y=16$$

Tāpiri \frac{8y}{3} ki te 4y.

$$\frac{20}{3}y-4=16$$

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

$$\frac{20}{3}y=20$$

Whakawehea ngā taha e rua o te whārite ki te \frac{20}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=3$$

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

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

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

$$x=4-2$$

Tāpiri -2 ki te 4.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=3$$

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.

$$3x-4y=-6,2x+4y=16$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&-4\\2&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-6\\16\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}3&-4\\2&4\end{matrix}\right))\left(\begin{matrix}3&-4\\2&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-4\\2&4\end{matrix}\right))\left(\begin{matrix}-6\\16\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-4\\2&4\end{matrix}\right))\left(\begin{matrix}-6\\16\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-4\\2&4\end{matrix}\right))\left(\begin{matrix}-6\\16\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{3\times 4-\left(-4\times 2\right)}&-\frac{-4}{3\times 4-\left(-4\times 2\right)}\\-\frac{2}{3\times 4-\left(-4\times 2\right)}&\frac{3}{3\times 4-\left(-4\times 2\right)}\end{matrix}\right)\left(\begin{matrix}-6\\16\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{5}&\frac{1}{5}\\-\frac{1}{10}&\frac{3}{20}\end{matrix}\right)\left(\begin{matrix}-6\\16\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{5}\left(-6\right)+\frac{1}{5}\times 16\\-\frac{1}{10}\left(-6\right)+\frac{3}{20}\times 16\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=2,y=3$$

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.

$$3x-4y=-6,2x+4y=16$$

Kia ōrite ai a 3x 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 3.

$$2\times 3x+2\left(-4\right)y=2\left(-6\right),3\times 2x+3\times 4y=3\times 16$$

Whakarūnātia.

$$6x-8y=-12,6x+12y=48$$

Me tango 6x+12y=48 mai i 6x-8y=-12 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x-8y-12y=-12-48$$

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.

$$-8y-12y=-12-48$$

Tāpiri -8y ki te -12y.

$$-20y=-12-48$$

Tāpiri -12 ki te -48.

$$-20y=-60$$

Whakawehea ngā taha e rua ki te -20.

$$y=3$$

Whakaurua te 3 mō y ki 2x+4y=16. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x+4\times 3=16$$

Whakareatia 4 ki te 3.

$$2x+12=16$$

Me tango 12 mai i ngā taha e rua o te whārite.

$$2x=4$$

Whakawehea ngā taha e rua ki te 2.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=3$$