Whakaoti i te {l}{2x-5y=-3}{-4x+y=-3}

$$\left\{ \begin{array} { l } { 2 x - 5 y = - 3 } \\ { - 4 x + y = - 3 } \end{array} \right.$$

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.

$$2x-5y=-3,-4x+y=-3$$

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

$$2x-5y=-3$$

Me tāpiri 5y ki ngā taha e rua o te whārite.

$$2x=5y-3$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(5y-3\right)$$

Whakareatia \frac{1}{2} ki te 5y-3.

$$x=\frac{5}{2}y-\frac{3}{2}$$

Whakakapia te \frac{5y-3}{2} mō te x ki tērā atu whārite, -4x+y=-3.

$$-4\left(\frac{5}{2}y-\frac{3}{2}\right)+y=-3$$

Whakareatia -4 ki te \frac{5y-3}{2}.

$$-10y+6+y=-3$$

Tāpiri -10y ki te y.

$$-9y+6=-3$$

Me tango 6 mai i ngā taha e rua o te whārite.

$$-9y=-9$$

Whakawehea ngā taha e rua ki te -9.

$$y=1$$

Whakaurua te 1 mō y ki x=\frac{5}{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.

$$x=\frac{5-3}{2}$$

Tāpiri -\frac{3}{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.

$$x=1$$

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.

$$2x-5y=-3,-4x+y=-3$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&-5\\-4&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-3\\-3\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}2&-5\\-4&1\end{matrix}\right))\left(\begin{matrix}2&-5\\-4&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-5\\-4&1\end{matrix}\right))\left(\begin{matrix}-3\\-3\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-5\\-4&1\end{matrix}\right))\left(\begin{matrix}-3\\-3\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-5\\-4&1\end{matrix}\right))\left(\begin{matrix}-3\\-3\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2-\left(-5\left(-4\right)\right)}&-\frac{-5}{2-\left(-5\left(-4\right)\right)}\\-\frac{-4}{2-\left(-5\left(-4\right)\right)}&\frac{2}{2-\left(-5\left(-4\right)\right)}\end{matrix}\right)\left(\begin{matrix}-3\\-3\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{18}&-\frac{5}{18}\\-\frac{2}{9}&-\frac{1}{9}\end{matrix}\right)\left(\begin{matrix}-3\\-3\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{18}\left(-3\right)-\frac{5}{18}\left(-3\right)\\-\frac{2}{9}\left(-3\right)-\frac{1}{9}\left(-3\right)\end{matrix}\right)$$

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.

$$2x-5y=-3,-4x+y=-3$$

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.

$$-4\times 2x-4\left(-5\right)y=-4\left(-3\right),2\left(-4\right)x+2y=2\left(-3\right)$$

Whakarūnātia.

$$-8x+20y=12,-8x+2y=-6$$

Me tango -8x+2y=-6 mai i -8x+20y=12 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-8x+8x+20y-2y=12+6$$

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.

$$20y-2y=12+6$$

Tāpiri 20y ki te -2y.

$$18y=12+6$$

Tāpiri 12 ki te 6.

$$18y=18$$

Whakawehea ngā taha e rua ki te 18.

$$y=1$$

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 tango 1 mai i 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.

$$x=1,y=1$$