Whakaoti i te {l}{5x-y=3}{-2x+4y=12}

$$\left. \begin{array} { l } { 5 x - y = 3 } \\ { - 2 x + 4 y = 12 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{4}{3} = 1\frac{1}{3} \approx 1.333333333$$

$$y = \frac{11}{3} = 3\frac{2}{3} \approx 3.666666667$$

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.

$$5x-y=3,-2x+4y=12$$

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

$$5x-y=3$$

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

$$5x=y+3$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\left(y+3\right)$$

Whakareatia \frac{1}{5} ki te y+3.

$$x=\frac{1}{5}y+\frac{3}{5}$$

Whakakapia te \frac{3+y}{5} mō te x ki tērā atu whārite, -2x+4y=12.

$$-2\left(\frac{1}{5}y+\frac{3}{5}\right)+4y=12$$

Whakareatia -2 ki te \frac{3+y}{5}.

$$-\frac{2}{5}y-\frac{6}{5}+4y=12$$

Tāpiri -\frac{2y}{5} ki te 4y.

$$\frac{18}{5}y-\frac{6}{5}=12$$

Me tāpiri \frac{6}{5} ki ngā taha e rua o te whārite.

$$\frac{18}{5}y=\frac{66}{5}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{18}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{11}{3}$$

Whakaurua te \frac{11}{3} mō y ki x=\frac{1}{5}y+\frac{3}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{1}{5}\times \frac{11}{3}+\frac{3}{5}$$

Whakareatia \frac{1}{5} ki te \frac{11}{3} 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.

$$x=\frac{11}{15}+\frac{3}{5}$$

Tāpiri \frac{3}{5} ki te \frac{11}{15} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=\frac{4}{3}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{4}{3},y=\frac{11}{3}$$

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.

$$5x-y=3,-2x+4y=12$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&-1\\-2&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3\\12\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&-1\\-2&4\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&-1\\-2&4\end{matrix}\right))\left(\begin{matrix}5&-1\\-2&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-1\\-2&4\end{matrix}\right))\left(\begin{matrix}3\\12\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-1\\-2&4\end{matrix}\right))\left(\begin{matrix}3\\12\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-1\\-2&4\end{matrix}\right))\left(\begin{matrix}3\\12\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{5\times 4-\left(-\left(-2\right)\right)}&-\frac{-1}{5\times 4-\left(-\left(-2\right)\right)}\\-\frac{-2}{5\times 4-\left(-\left(-2\right)\right)}&\frac{5}{5\times 4-\left(-\left(-2\right)\right)}\end{matrix}\right)\left(\begin{matrix}3\\12\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{9}&\frac{1}{18}\\\frac{1}{9}&\frac{5}{18}\end{matrix}\right)\left(\begin{matrix}3\\12\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{9}\times 3+\frac{1}{18}\times 12\\\frac{1}{9}\times 3+\frac{5}{18}\times 12\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{3}\\\frac{11}{3}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{4}{3},y=\frac{11}{3}$$

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.

$$5x-y=3,-2x+4y=12$$

Kia ōrite ai a 5x 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 5.

$$-2\times 5x-2\left(-1\right)y=-2\times 3,5\left(-2\right)x+5\times 4y=5\times 12$$

Whakarūnātia.

$$-10x+2y=-6,-10x+20y=60$$

Me tango -10x+20y=60 mai i -10x+2y=-6 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-10x+10x+2y-20y=-6-60$$

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.

$$2y-20y=-6-60$$

Tāpiri 2y ki te -20y.

$$-18y=-6-60$$

Tāpiri -6 ki te -60.

$$-18y=-66$$

Whakawehea ngā taha e rua ki te -18.

$$y=\frac{11}{3}$$

Whakaurua te \frac{11}{3} mō y ki -2x+4y=12. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-2x+4\times \frac{11}{3}=12$$

Whakareatia 4 ki te \frac{11}{3}.

$$-2x+\frac{44}{3}=12$$

Me tango \frac{44}{3} mai i ngā taha e rua o te whārite.

$$-2x=-\frac{8}{3}$$

Whakawehea ngā taha e rua ki te -2.

$$x=\frac{4}{3}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{4}{3},y=\frac{11}{3}$$