Whakaoti i te {l}{2x-y=0}{5x-2y=1}

$$\left. \begin{array} { l } { 2 x - y = 0 } \\ { 5 x - 2 y = 1 } \end{array} \right.$$

Whakaoti mō x, y

$$x=1$$

$$y=2$$

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.

$$2x-y=0,5x-2y=1$$

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

$$2x-y=0$$

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

$$2x=y$$

Whakawehea ngā taha e rua ki te 2.

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

Whakakapia te \frac{y}{2} mō te x ki tērā atu whārite, 5x-2y=1.

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

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

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

Tāpiri \frac{5y}{2} ki te -2y.

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

Me whakarea ngā taha e rua ki te 2.

$$y=2$$

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

$$x=\frac{1}{2}\times 2$$

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

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=2$$

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.

$$2x-y=0,5x-2y=1$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&-1\\5&-2\end{matrix}\right).

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Tangohia ngā huānga poukapa x me y.

$$x=1,y=2$$

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.

$$2x-y=0,5x-2y=1$$

Kia ōrite ai a 2x me 5x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 5 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

$$5\times 2x+5\left(-1\right)y=0,2\times 5x+2\left(-2\right)y=2$$

Whakarūnātia.

$$10x-5y=0,10x-4y=2$$

Me tango 10x-4y=2 mai i 10x-5y=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$10x-10x-5y+4y=-2$$

Tāpiri 10x ki te -10x. Ka whakakore atu ngā kupu 10x me -10x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

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

Tāpiri -5y ki te 4y.

$$-y=-2$$

Whakawehea ngā taha e rua ki te -1.

$$y=2$$

Whakaurua te 2 mō y ki 5x-2y=1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$5x-2\times 2=1$$

Whakareatia -2 ki te 2.

$$5x-4=1$$

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

$$5x=5$$

Whakawehea ngā taha e rua ki te 5.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=2$$