Whakaoti i te {l}{-2x+y=-1}{4x-y=-3}

$$\left. \begin{array} { l } { - 2 x + y = - 1 } \\ { 4 x - y = - 3 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-2$$

$$y=-5$$

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.

$$-2x+y=-1,4x-y=-3$$

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

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

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

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

Whakawehea ngā taha e rua ki te -2.

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

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

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

Whakakapia te \frac{1+y}{2} mō te x ki tērā atu whārite, 4x-y=-3.

$$4\left(\frac{1}{2}y+\frac{1}{2}\right)-y=-3$$

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

$$2y+2-y=-3$$

Tāpiri 2y ki te -y.

$$y+2=-3$$

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

$$y=-5$$

Whakaurua te -5 mō y ki x=\frac{1}{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.

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

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

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

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.

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

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.

$$-2x+y=-1,4x-y=-3$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-2&1\\4&-1\end{matrix}\right).

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

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-2&1\\4&-1\end{matrix}\right))\left(\begin{matrix}-1\\-3\end{matrix}\right)$$

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

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

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.

$$-2x+y=-1,4x-y=-3$$

Kia ōrite ai a -2x me 4x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 4 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te -2.

$$4\left(-2\right)x+4y=4\left(-1\right),-2\times 4x-2\left(-1\right)y=-2\left(-3\right)$$

Whakarūnātia.

$$-8x+4y=-4,-8x+2y=6$$

Me tango -8x+2y=6 mai i -8x+4y=-4 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-8x+8x+4y-2y=-4-6$$

Tāpiri -8x ki te 8x. Ka whakakore atu ngā kupu -8x me 8x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$4y-2y=-4-6$$

Tāpiri 4y ki te -2y.

$$2y=-4-6$$

Tāpiri -4 ki te -6.

$$2y=-10$$

Whakawehea ngā taha e rua ki te 2.

$$y=-5$$

Whakaurua te -5 mō y ki 4x-y=-3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x-\left(-5\right)=-3$$

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

$$4x=-8$$

Whakawehea ngā taha e rua ki te 4.

$$x=-2$$

Kua oti te pūnaha te whakatau.

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