Whakaoti i te {c}{8x-4y=-1}{x+y=1}

$$\left\{ \begin{array} { c } { 8 x - 4 y = - 1 } \\ { x + y = 1 } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{1}{4}=0.25$$

$$y=\frac{3}{4}=0.75$$

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.

$$8x-4y=-1,x+y=1$$

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

$$8x-4y=-1$$

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

$$8x=4y-1$$

Whakawehea ngā taha e rua ki te 8.

$$x=\frac{1}{8}\left(4y-1\right)$$

Whakareatia \frac{1}{8} ki te 4y-1.

$$x=\frac{1}{2}y-\frac{1}{8}$$

Whakakapia te \frac{y}{2}-\frac{1}{8} mō te x ki tērā atu whārite, x+y=1.

$$\frac{1}{2}y-\frac{1}{8}+y=1$$

Tāpiri \frac{y}{2} ki te y.

$$\frac{3}{2}y-\frac{1}{8}=1$$

Me tāpiri \frac{1}{8} ki ngā taha e rua o te whārite.

$$\frac{3}{2}y=\frac{9}{8}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{3}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

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

Whakaurua te \frac{3}{4} mō y ki x=\frac{1}{2}y-\frac{1}{8}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{1}{2}\times \frac{3}{4}-\frac{1}{8}$$

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

$$x=\frac{3-1}{8}$$

Tāpiri -\frac{1}{8} ki te \frac{3}{8} 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.

$$x=\frac{1}{4}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{1}{4},y=\frac{3}{4}$$

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.

$$8x-4y=-1,x+y=1$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}8&-4\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-1\\1\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}8&-4\\1&1\end{matrix}\right).

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

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

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

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

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

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{8-\left(-4\right)}&-\frac{-4}{8-\left(-4\right)}\\-\frac{1}{8-\left(-4\right)}&\frac{8}{8-\left(-4\right)}\end{matrix}\right)\left(\begin{matrix}-1\\1\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}\\\frac{3}{4}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{1}{4},y=\frac{3}{4}$$

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.

$$8x-4y=-1,x+y=1$$

Kia ōrite ai a 8x me x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 1 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 8.

$$8x-4y=-1,8x+8y=8$$

Me tango 8x+8y=8 mai i 8x-4y=-1 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$8x-8x-4y-8y=-1-8$$

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.

$$-4y-8y=-1-8$$

Tāpiri -4y ki te -8y.

$$-12y=-1-8$$

Tāpiri -1 ki te -8.

$$-12y=-9$$

Whakawehea ngā taha e rua ki te -12.

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

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

$$x+\frac{3}{4}=1$$

Me tango \frac{3}{4} mai i ngā taha e rua o te whārite.

$$x=\frac{1}{4}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{1}{4},y=\frac{3}{4}$$