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

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

Whakaoti mō x, y

$$x = -\frac{5}{2} = -2\frac{1}{2} = -2.5$$

$$y = \frac{5}{2} = 2\frac{1}{2} = 2.5$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Me tāpiri te y ki ngā taha e rua.

$$x+y=0$$

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.

$$2x+4y=5,x+y=0$$

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

$$2x+4y=5$$

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

$$2x=-4y+5$$

Whakawehea ngā taha e rua ki te 2.

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

Whakareatia \frac{1}{2} ki te -4y+5.

$$x=-2y+\frac{5}{2}$$

Whakakapia te -2y+\frac{5}{2} mō te x ki tērā atu whārite, x+y=0.

$$-2y+\frac{5}{2}+y=0$$

Tāpiri -2y ki te y.

$$-y+\frac{5}{2}=0$$

Me tango \frac{5}{2} mai i ngā taha e rua o te whārite.

$$-y=-\frac{5}{2}$$

Whakawehea ngā taha e rua ki te -1.

$$y=\frac{5}{2}$$

Whakaurua te \frac{5}{2} mō y ki x=-2y+\frac{5}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-2\times \frac{5}{2}+\frac{5}{2}$$

Whakareatia -2 ki te \frac{5}{2}.

$$x=-5+\frac{5}{2}$$

Tāpiri \frac{5}{2} ki te -5.

$$x=-\frac{5}{2}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{5}{2},y=\frac{5}{2}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Me tāpiri te y ki ngā taha e rua.

$$x+y=0$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$2x+4y=5,x+y=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{2}\\\frac{5}{2}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{5}{2},y=\frac{5}{2}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Me tāpiri te y ki ngā taha e rua.

$$x+y=0$$

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.

$$2x+4y=5,x+y=0$$

Kia ōrite ai a 2x 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 2.

$$2x+4y=5,2x+2y=0$$

Me tango 2x+2y=0 mai i 2x+4y=5 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2x-2x+4y-2y=5$$

Tāpiri 2x ki te -2x. Ka whakakore atu ngā kupu 2x me -2x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$4y-2y=5$$

Tāpiri 4y ki te -2y.

$$2y=5$$

Whakawehea ngā taha e rua ki te 2.

$$y=\frac{5}{2}$$

Whakaurua te \frac{5}{2} mō y ki x+y=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x+\frac{5}{2}=0$$

Me tango \frac{5}{2} mai i ngā taha e rua o te whārite.

$$x=-\frac{5}{2}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{5}{2},y=\frac{5}{2}$$