Whakaoti i te {l}{2x+y=-2}{4x+5y=8}

$$\left\{ \begin{array} { l } { 2 x + y = - 2 } \\ { 4 x + 5 y = 8 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-3$$

$$y=4$$

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.

$$2x+y=-2,4x+5y=8$$

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

$$2x+y=-2$$

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

$$2x=-y-2$$

Whakawehea ngā taha e rua ki te 2.

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

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

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

Whakakapia te -\frac{y}{2}-1 mō te x ki tērā atu whārite, 4x+5y=8.

$$4\left(-\frac{1}{2}y-1\right)+5y=8$$

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

$$-2y-4+5y=8$$

Tāpiri -2y ki te 5y.

$$3y-4=8$$

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

$$3y=12$$

Whakawehea ngā taha e rua ki te 3.

$$y=4$$

Whakaurua te 4 mō y ki x=-\frac{1}{2}y-1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{2}\times 4-1$$

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

$$x=-2-1$$

Tāpiri -1 ki te -2.

$$x=-3$$

Kua oti te pūnaha te whakatau.

$$x=-3,y=4$$

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.

$$2x+y=-2,4x+5y=8$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&1\\4&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-2\\8\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&1\\4&5\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&1\\4&5\end{matrix}\right))\left(\begin{matrix}2&1\\4&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\4&5\end{matrix}\right))\left(\begin{matrix}-2\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\4&5\end{matrix}\right))\left(\begin{matrix}-2\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\4&5\end{matrix}\right))\left(\begin{matrix}-2\\8\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{2\times 5-4}&-\frac{1}{2\times 5-4}\\-\frac{4}{2\times 5-4}&\frac{2}{2\times 5-4}\end{matrix}\right)\left(\begin{matrix}-2\\8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{6}&-\frac{1}{6}\\-\frac{2}{3}&\frac{1}{3}\end{matrix}\right)\left(\begin{matrix}-2\\8\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{6}\left(-2\right)-\frac{1}{6}\times 8\\-\frac{2}{3}\left(-2\right)+\frac{1}{3}\times 8\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=-3,y=4$$

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.

$$2x+y=-2,4x+5y=8$$

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.

$$4\times 2x+4y=4\left(-2\right),2\times 4x+2\times 5y=2\times 8$$

Whakarūnātia.

$$8x+4y=-8,8x+10y=16$$

Me tango 8x+10y=16 mai i 8x+4y=-8 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$8x-8x+4y-10y=-8-16$$

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.

$$4y-10y=-8-16$$

Tāpiri 4y ki te -10y.

$$-6y=-8-16$$

Tāpiri -8 ki te -16.

$$-6y=-24$$

Whakawehea ngā taha e rua ki te -6.

$$y=4$$

Whakaurua te 4 mō y ki 4x+5y=8. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x+5\times 4=8$$

Whakareatia 5 ki te 4.

$$4x+20=8$$

Me tango 20 mai i ngā taha e rua o te whārite.

$$4x=-12$$

Whakawehea ngā taha e rua ki te 4.

$$x=-3$$

Kua oti te pūnaha te whakatau.

$$x=-3,y=4$$