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

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

Whakaoti mō x, y

$$x=1$$

$$y=-1$$

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.

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

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

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

Me tango 3y mai i ngā taha e rua o te whārite.

$$2x=-3y-1$$

Whakawehea ngā taha e rua ki te 2.

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

Whakareatia \frac{1}{2} ki te -3y-1.

$$x=-\frac{3}{2}y-\frac{1}{2}$$

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

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

Whakareatia 4 ki te \frac{-3y-1}{2}.

$$-6y-2+y=3$$

Tāpiri -6y ki te y.

$$-5y-2=3$$

Me tāpiri 2 ki ngā taha e rua o te whārite.

$$-5y=5$$

Whakawehea ngā taha e rua ki te -5.

$$y=-1$$

Whakaurua te -1 mō y ki x=-\frac{3}{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.

$$x=-\frac{3}{2}\left(-1\right)-\frac{1}{2}$$

Whakareatia -\frac{3}{2} ki te -1.

$$x=\frac{3-1}{2}$$

Tāpiri -\frac{1}{2} ki te \frac{3}{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.

$$x=1$$

Kua oti te pūnaha te whakatau.

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

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&3\\4&1\end{matrix}\right).

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{10}&\frac{3}{10}\\\frac{2}{5}&-\frac{1}{5}\end{matrix}\right)\left(\begin{matrix}-1\\3\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{10}\left(-1\right)+\frac{3}{10}\times 3\\\frac{2}{5}\left(-1\right)-\frac{1}{5}\times 3\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

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

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.

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

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.

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

Whakarūnātia.

$$8x+12y=-4,8x+2y=6$$

Me tango 8x+2y=6 mai i 8x+12y=-4 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$8x-8x+12y-2y=-4-6$$

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.

$$12y-2y=-4-6$$

Tāpiri 12y ki te -2y.

$$10y=-4-6$$

Tāpiri -4 ki te -6.

$$10y=-10$$

Whakawehea ngā taha e rua ki te 10.

$$y=-1$$

Whakaurua te -1 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.

$$4x-1=3$$

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

$$4x=4$$

Whakawehea ngā taha e rua ki te 4.

$$x=1$$

Kua oti te pūnaha te whakatau.

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