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

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

Whakaoti mō x, y

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

$$y=0$$

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.

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

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

$$2x+y=5$$

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

$$2x=-y+5$$

Whakawehea ngā taha e rua ki te 2.

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

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

$$x=-\frac{1}{2}y+\frac{5}{2}$$

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

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

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

$$-y+5-y=5$$

Tāpiri -y ki te -y.

$$-2y+5=5$$

Me tango 5 mai i ngā taha e rua o te whārite.

$$-2y=0$$

Whakawehea ngā taha e rua ki te -2.

$$y=0$$

Whakaurua te 0 mō y ki x=-\frac{1}{2}y+\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.

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

Kua oti te pūnaha te whakatau.

$$x=\frac{5}{2},y=0$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=\frac{5}{2},y=0$$

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.

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

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

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

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.

$$y+y=5-5$$

Tāpiri y ki te y.

$$2y=5-5$$

Tāpiri 5 ki te -5.

$$2y=0$$

Whakawehea ngā taha e rua ki te 2.

$$y=0$$

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

$$2x=5$$

Whakawehea ngā taha e rua ki te 2.

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

Kua oti te pūnaha te whakatau.

$$x=\frac{5}{2},y=0$$