Whakaoti i te {l}{3x-y-2=0}{2x+y-8=0}

$$\left. \begin{array} { l } { 3 x - y - 2 = 0 } \\ { 2 x + y - 8 = 0 } \end{array} \right.$$

Whakaoti mō x, y

$$x=2$$

$$y=4$$

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.

$$3x-y-2=0,2x+y-8=0$$

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

$$3x-y-2=0$$

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

$$3x-y=2$$

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

$$3x=y+2$$

Whakawehea ngā taha e rua ki te 3.

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

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

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

Whakakapia te \frac{2+y}{3} mō te x ki tērā atu whārite, 2x+y-8=0.

$$2\left(\frac{1}{3}y+\frac{2}{3}\right)+y-8=0$$

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

$$\frac{2}{3}y+\frac{4}{3}+y-8=0$$

Tāpiri \frac{2y}{3} ki te y.

$$\frac{5}{3}y+\frac{4}{3}-8=0$$

Tāpiri \frac{4}{3} ki te -8.

$$\frac{5}{3}y-\frac{20}{3}=0$$

Me tāpiri \frac{20}{3} ki ngā taha e rua o te whārite.

$$\frac{5}{3}y=\frac{20}{3}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{5}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=4$$

Whakaurua te 4 mō y ki x=\frac{1}{3}y+\frac{2}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{1}{3}\times 4+\frac{2}{3}$$

Whakareatia \frac{1}{3} ki te 4.

$$x=\frac{4+2}{3}$$

Tāpiri \frac{2}{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.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=4$$

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.

$$3x-y-2=0,2x+y-8=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&-1\\2&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2\\8\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}3&-1\\2&1\end{matrix}\right))\left(\begin{matrix}3&-1\\2&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-1\\2&1\end{matrix}\right))\left(\begin{matrix}2\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-1\\2&1\end{matrix}\right))\left(\begin{matrix}2\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-1\\2&1\end{matrix}\right))\left(\begin{matrix}2\\8\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3-\left(-2\right)}&-\frac{-1}{3-\left(-2\right)}\\-\frac{2}{3-\left(-2\right)}&\frac{3}{3-\left(-2\right)}\end{matrix}\right)\left(\begin{matrix}2\\8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{5}&\frac{1}{5}\\-\frac{2}{5}&\frac{3}{5}\end{matrix}\right)\left(\begin{matrix}2\\8\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{5}\times 2+\frac{1}{5}\times 8\\-\frac{2}{5}\times 2+\frac{3}{5}\times 8\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=2,y=4$$

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.

$$3x-y-2=0,2x+y-8=0$$

Kia ōrite ai a 3x me 2x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$2\times 3x+2\left(-1\right)y+2\left(-2\right)=0,3\times 2x+3y+3\left(-8\right)=0$$

Whakarūnātia.

$$6x-2y-4=0,6x+3y-24=0$$

Me tango 6x+3y-24=0 mai i 6x-2y-4=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x-2y-3y-4+24=0$$

Tāpiri 6x ki te -6x. Ka whakakore atu ngā kupu 6x me -6x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-2y-3y-4+24=0$$

Tāpiri -2y ki te -3y.

$$-5y-4+24=0$$

Tāpiri -4 ki te 24.

$$-5y+20=0$$

Me tango 20 mai i ngā taha e rua o te whārite.

$$-5y=-20$$

Whakawehea ngā taha e rua ki te -5.

$$y=4$$

Whakaurua te 4 mō y ki 2x+y-8=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x+4-8=0$$

Tāpiri 4 ki te -8.

$$2x-4=0$$

Me tāpiri 4 ki ngā taha e rua o te whārite.

$$2x=4$$

Whakawehea ngā taha e rua ki te 2.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=4$$