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

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

Whakaoti mō x, y

$$x = \frac{23}{20} = 1\frac{3}{20} = 1.15$$

$$y=-\frac{1}{10}=-0.1$$

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.

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

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

$$2x+3y=2$$

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

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

Whakawehea ngā taha e rua ki te 2.

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

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

$$x=-\frac{3}{2}y+1$$

Whakakapia te -\frac{3y}{2}+1 mō te x ki tērā atu whārite, 4x+16y=3.

$$4\left(-\frac{3}{2}y+1\right)+16y=3$$

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

$$-6y+4+16y=3$$

Tāpiri -6y ki te 16y.

$$10y+4=3$$

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

$$10y=-1$$

Whakawehea ngā taha e rua ki te 10.

$$y=-\frac{1}{10}$$

Whakaurua te -\frac{1}{10} mō y ki x=-\frac{3}{2}y+1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{3}{2}\left(-\frac{1}{10}\right)+1$$

Whakareatia -\frac{3}{2} ki te -\frac{1}{10} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=\frac{3}{20}+1$$

Tāpiri 1 ki te \frac{3}{20}.

$$x=\frac{23}{20}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{23}{20},y=-\frac{1}{10}$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&3\\4&16\end{matrix}\right).

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{5}&-\frac{3}{20}\\-\frac{1}{5}&\frac{1}{10}\end{matrix}\right)\left(\begin{matrix}2\\3\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{5}\times 2-\frac{3}{20}\times 3\\-\frac{1}{5}\times 2+\frac{1}{10}\times 3\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{23}{20}\\-\frac{1}{10}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{23}{20},y=-\frac{1}{10}$$

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.

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

Kia ōrite ai a 2x me 4x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 4 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

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

Whakarūnātia.

$$8x+12y=8,8x+32y=6$$

Me tango 8x+32y=6 mai i 8x+12y=8 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$8x-8x+12y-32y=8-6$$

Tāpiri 8x ki te -8x. Ka whakakore atu ngā kupu 8x me -8x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$12y-32y=8-6$$

Tāpiri 12y ki te -32y.

$$-20y=8-6$$

Tāpiri 8 ki te -6.

$$-20y=2$$

Whakawehea ngā taha e rua ki te -20.

$$y=-\frac{1}{10}$$

Whakaurua te -\frac{1}{10} mō y ki 4x+16y=3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x+16\left(-\frac{1}{10}\right)=3$$

Whakareatia 16 ki te -\frac{1}{10}.

$$4x-\frac{8}{5}=3$$

Me tāpiri \frac{8}{5} ki ngā taha e rua o te whārite.

$$4x=\frac{23}{5}$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{23}{20}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{23}{20},y=-\frac{1}{10}$$