Whakaoti i te {l}{3x-2y=5}{-3x+4y=-9}

$$\left\{ \begin{array} { l } { 3 x - 2 y = 5 } \\ { - 3 x + 4 y = - 9 } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{1}{3}\approx 0.333333333$$

$$y=-2$$

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.

$$3x-2y=5,-3x+4y=-9$$

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

$$3x-2y=5$$

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

$$3x=2y+5$$

Whakawehea ngā taha e rua ki te 3.

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

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

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

Whakakapia te \frac{2y+5}{3} mō te x ki tērā atu whārite, -3x+4y=-9.

$$-3\left(\frac{2}{3}y+\frac{5}{3}\right)+4y=-9$$

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

$$-2y-5+4y=-9$$

Tāpiri -2y ki te 4y.

$$2y-5=-9$$

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

$$2y=-4$$

Whakawehea ngā taha e rua ki te 2.

$$y=-2$$

Whakaurua te -2 mō y ki x=\frac{2}{3}y+\frac{5}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

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

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

$$x=\frac{-4+5}{3}$$

Tāpiri \frac{5}{3} ki te -\frac{4}{3} 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.

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

Kua oti te pūnaha te whakatau.

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

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.

$$3x-2y=5,-3x+4y=-9$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&-2\\-3&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\\-9\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}3&-2\\-3&4\end{matrix}\right))\left(\begin{matrix}3&-2\\-3&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-2\\-3&4\end{matrix}\right))\left(\begin{matrix}5\\-9\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-2\\-3&4\end{matrix}\right))\left(\begin{matrix}5\\-9\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-2\\-3&4\end{matrix}\right))\left(\begin{matrix}5\\-9\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{3\times 4-\left(-2\left(-3\right)\right)}&-\frac{-2}{3\times 4-\left(-2\left(-3\right)\right)}\\-\frac{-3}{3\times 4-\left(-2\left(-3\right)\right)}&\frac{3}{3\times 4-\left(-2\left(-3\right)\right)}\end{matrix}\right)\left(\begin{matrix}5\\-9\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

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

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.

$$3x-2y=5,-3x+4y=-9$$

Kia ōrite ai a 3x me -3x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -3 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$-3\times 3x-3\left(-2\right)y=-3\times 5,3\left(-3\right)x+3\times 4y=3\left(-9\right)$$

Whakarūnātia.

$$-9x+6y=-15,-9x+12y=-27$$

Me tango -9x+12y=-27 mai i -9x+6y=-15 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-9x+9x+6y-12y=-15+27$$

Tāpiri -9x ki te 9x. Ka whakakore atu ngā kupu -9x me 9x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$6y-12y=-15+27$$

Tāpiri 6y ki te -12y.

$$-6y=-15+27$$

Tāpiri -15 ki te 27.

$$-6y=12$$

Whakawehea ngā taha e rua ki te -6.

$$y=-2$$

Whakaurua te -2 mō y ki -3x+4y=-9. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-3x+4\left(-2\right)=-9$$

Whakareatia 4 ki te -2.

$$-3x-8=-9$$

Me tāpiri 8 ki ngā taha e rua o te whārite.

$$-3x=-1$$

Whakawehea ngā taha e rua ki te -3.

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

Kua oti te pūnaha te whakatau.

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