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

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

Whakaoti mō x, y

$$x=-2$$

$$y=1$$

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.

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

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

$$2x+5y=1$$

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

$$2x=-5y+1$$

Whakawehea ngā taha e rua ki te 2.

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

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

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

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

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

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

$$5y-1+y=5$$

Tāpiri 5y ki te y.

$$6y-1=5$$

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

$$6y=6$$

Whakawehea ngā taha e rua ki te 6.

$$y=1$$

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

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

Tāpiri \frac{1}{2} ki te -\frac{5}{2} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=-2$$

Kua oti te pūnaha te whakatau.

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

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{12}&-\frac{5}{12}\\\frac{1}{6}&\frac{1}{6}\end{matrix}\right)\left(\begin{matrix}1\\5\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{12}-\frac{5}{12}\times 5\\\frac{1}{6}+\frac{1}{6}\times 5\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

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

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.

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

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

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

Whakarūnātia.

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

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

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

Tāpiri -4x ki te 4x. Ka whakakore atu ngā kupu -4x me 4x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-10y-2y=-2-10$$

Tāpiri -10y ki te -2y.

$$-12y=-2-10$$

Tāpiri -2 ki te -10.

$$-12y=-12$$

Whakawehea ngā taha e rua ki te -12.

$$y=1$$

Whakaurua te 1 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.

$$-2x+1=5$$

Me tango 1 mai i ngā taha e rua o te whārite.

$$-2x=4$$

Whakawehea ngā taha e rua ki te -2.

$$x=-2$$

Kua oti te pūnaha te whakatau.

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