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

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

Whakaoti mō x, y

$$x=0$$

$$y=-2$$

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.

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

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

$$-x-2y=4$$

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

$$-x=2y+4$$

Whakawehea ngā taha e rua ki te -1.

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

Whakareatia -1 ki te 4+2y.

$$x=-2y-4$$

Whakakapia te -2y-4 mō te x ki tērā atu whārite, 3x-y=2.

$$3\left(-2y-4\right)-y=2$$

Whakareatia 3 ki te -2y-4.

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

Tāpiri -6y ki te -y.

$$-7y-12=2$$

Me tāpiri 12 ki ngā taha e rua o te whārite.

$$-7y=14$$

Whakawehea ngā taha e rua ki te -7.

$$y=-2$$

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

$$x=-2\left(-2\right)-4$$

Whakareatia -2 ki te -2.

$$x=4-4$$

Tāpiri -4 ki te 4.

$$x=0$$

Kua oti te pūnaha te whakatau.

$$x=0,y=-2$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{7}&\frac{2}{7}\\-\frac{3}{7}&-\frac{1}{7}\end{matrix}\right)\left(\begin{matrix}4\\2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{7}\times 4+\frac{2}{7}\times 2\\-\frac{3}{7}\times 4-\frac{1}{7}\times 2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}0\\-2\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=0,y=-2$$

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.

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

Kia ōrite ai a -x 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 -1.

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

Whakarūnātia.

$$-3x-6y=12,-3x+y=-2$$

Me tango -3x+y=-2 mai i -3x-6y=12 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-3x+3x-6y-y=12+2$$

Tāpiri -3x ki te 3x. Ka whakakore atu ngā kupu -3x me 3x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

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

Tāpiri -6y ki te -y.

$$-7y=12+2$$

Tāpiri 12 ki te 2.

$$-7y=14$$

Whakawehea ngā taha e rua ki te -7.

$$y=-2$$

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

$$3x-\left(-2\right)=2$$

Me tango 2 mai i ngā taha e rua o te whārite.

$$3x=0$$

Whakawehea ngā taha e rua ki te 3.

$$x=0$$

Kua oti te pūnaha te whakatau.

$$x=0,y=-2$$