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

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

Whakaoti mō x, y

$$x=3$$

$$y=-2$$

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.

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

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

$$2x+y=4$$

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, x-y=5.

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

Tāpiri -\frac{y}{2} ki te -y.

$$-\frac{3}{2}y+2=5$$

Me tango 2 mai i ngā taha e rua o te whārite.

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

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.

$$y=-2$$

Whakaurua te -2 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.

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

Whakareatia -\frac{1}{2} ki te -2.

$$x=1+2$$

Tāpiri 2 ki te 1.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=-2$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=3,y=-2$$

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.

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

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.

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

Whakarūnātia.

$$2x+y=4,2x-2y=10$$

Me tango 2x-2y=10 mai i 2x+y=4 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2x-2x+y+2y=4-10$$

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.

$$y+2y=4-10$$

Tāpiri y ki te 2y.

$$3y=4-10$$

Tāpiri 4 ki te -10.

$$3y=-6$$

Whakawehea ngā taha e rua ki te 3.

$$y=-2$$

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

$$x-\left(-2\right)=5$$

Me tango 2 mai i ngā taha e rua o te whārite.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=-2$$