Whakaoti i te {c}{2x-y=-3}{4x-3y=3}

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

Whakaoti mō x, y

$$x=-6$$

$$y=-9$$

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.

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

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

$$2x-y=-3$$

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

$$2x=y-3$$

Whakawehea ngā taha e rua ki te 2.

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

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

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

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

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

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

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

Tāpiri 2y ki te -3y.

$$-y-6=3$$

Me tāpiri 6 ki ngā taha e rua o te whārite.

$$-y=9$$

Whakawehea ngā taha e rua ki te -1.

$$y=-9$$

Whakaurua te -9 mō y ki x=\frac{1}{2}y-\frac{3}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{1}{2}\left(-9\right)-\frac{3}{2}$$

Whakareatia \frac{1}{2} ki te -9.

$$x=\frac{-9-3}{2}$$

Tāpiri -\frac{3}{2} ki te -\frac{9}{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.

$$x=-6$$

Kua oti te pūnaha te whakatau.

$$x=-6,y=-9$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-6\\-9\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-6,y=-9$$

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.

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

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.

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

Whakarūnātia.

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

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

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

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.

$$-4y+6y=-12-6$$

Tāpiri -4y ki te 6y.

$$2y=-12-6$$

Tāpiri -12 ki te -6.

$$2y=-18$$

Whakawehea ngā taha e rua ki te 2.

$$y=-9$$

Whakaurua te -9 mō y ki 4x-3y=3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x-3\left(-9\right)=3$$

Whakareatia -3 ki te -9.

$$4x+27=3$$

Me tango 27 mai i ngā taha e rua o te whārite.

$$4x=-24$$

Whakawehea ngā taha e rua ki te 4.

$$x=-6$$

Kua oti te pūnaha te whakatau.

$$x=-6,y=-9$$