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

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

Whakaoti mō x, y

$$x = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

$$y=1$$

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.

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

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

$$2x+y=4$$

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

$$2x=-y+4$$

Whakawehea ngā taha e rua ki te 2.

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

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

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

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

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

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

$$-y+4-y=2$$

Tāpiri -y ki te -y.

$$-2y+4=2$$

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

$$-2y=-2$$

Whakawehea ngā taha e rua ki te -2.

$$y=1$$

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

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

Tāpiri 2 ki te -\frac{1}{2}.

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

Kua oti te pūnaha te whakatau.

$$x=\frac{3}{2},y=1$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te poukapa kōaro ko \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), kia tuhia anō ai te whārite poukapa hei rapanga whakarea poukapa.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=\frac{3}{2},y=1$$

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.

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

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

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

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.

$$y+y=4-2$$

Tāpiri y ki te y.

$$2y=4-2$$

Tāpiri 4 ki te -2.

$$2y=2$$

Whakawehea ngā taha e rua ki te 2.

$$y=1$$

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

$$2x-1=2$$

Me tāpiri 1 ki ngā taha e rua o te whārite.

$$2x=3$$

Whakawehea ngā taha e rua ki te 2.

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

Kua oti te pūnaha te whakatau.

$$x=\frac{3}{2},y=1$$