Whakaoti i te {l}{2x+4y=12}{5x-8y=16}

$$\left\{ \begin{array} { l } { 2 x + 4 y = 12 } \\ { 5 x - 8 y = 16 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{40}{9} = 4\frac{4}{9} \approx 4.444444444$$

$$y=\frac{7}{9}\approx 0.777777778$$

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.

$$2x+4y=12,5x-8y=16$$

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

$$2x+4y=12$$

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

$$2x=-4y+12$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(-4y+12\right)$$

Whakareatia \frac{1}{2} ki te -4y+12.

$$x=-2y+6$$

Whakakapia te -2y+6 mō te x ki tērā atu whārite, 5x-8y=16.

$$5\left(-2y+6\right)-8y=16$$

Whakareatia 5 ki te -2y+6.

$$-10y+30-8y=16$$

Tāpiri -10y ki te -8y.

$$-18y+30=16$$

Me tango 30 mai i ngā taha e rua o te whārite.

$$-18y=-14$$

Whakawehea ngā taha e rua ki te -18.

$$y=\frac{7}{9}$$

Whakaurua te \frac{7}{9} mō y ki x=-2y+6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-2\times \frac{7}{9}+6$$

Whakareatia -2 ki te \frac{7}{9}.

$$x=-\frac{14}{9}+6$$

Tāpiri 6 ki te -\frac{14}{9}.

$$x=\frac{40}{9}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{40}{9},y=\frac{7}{9}$$

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.

$$2x+4y=12,5x-8y=16$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&4\\5&-8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}12\\16\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&4\\5&-8\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&4\\5&-8\end{matrix}\right))\left(\begin{matrix}2&4\\5&-8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&4\\5&-8\end{matrix}\right))\left(\begin{matrix}12\\16\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&4\\5&-8\end{matrix}\right))\left(\begin{matrix}12\\16\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&4\\5&-8\end{matrix}\right))\left(\begin{matrix}12\\16\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{8}{2\left(-8\right)-4\times 5}&-\frac{4}{2\left(-8\right)-4\times 5}\\-\frac{5}{2\left(-8\right)-4\times 5}&\frac{2}{2\left(-8\right)-4\times 5}\end{matrix}\right)\left(\begin{matrix}12\\16\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{9}&\frac{1}{9}\\\frac{5}{36}&-\frac{1}{18}\end{matrix}\right)\left(\begin{matrix}12\\16\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{9}\times 12+\frac{1}{9}\times 16\\\frac{5}{36}\times 12-\frac{1}{18}\times 16\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{40}{9}\\\frac{7}{9}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{40}{9},y=\frac{7}{9}$$

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.

$$2x+4y=12,5x-8y=16$$

Kia ōrite ai a 2x me 5x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 5 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

$$5\times 2x+5\times 4y=5\times 12,2\times 5x+2\left(-8\right)y=2\times 16$$

Whakarūnātia.

$$10x+20y=60,10x-16y=32$$

Me tango 10x-16y=32 mai i 10x+20y=60 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$10x-10x+20y+16y=60-32$$

Tāpiri 10x ki te -10x. Ka whakakore atu ngā kupu 10x me -10x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$20y+16y=60-32$$

Tāpiri 20y ki te 16y.

$$36y=60-32$$

Tāpiri 60 ki te -32.

$$36y=28$$

Whakawehea ngā taha e rua ki te 36.

$$y=\frac{7}{9}$$

Whakaurua te \frac{7}{9} mō y ki 5x-8y=16. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$5x-8\times \frac{7}{9}=16$$

Whakareatia -8 ki te \frac{7}{9}.

$$5x-\frac{56}{9}=16$$

Me tāpiri \frac{56}{9} ki ngā taha e rua o te whārite.

$$5x=\frac{200}{9}$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{40}{9}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{40}{9},y=\frac{7}{9}$$