Whakaoti i te {l}{8x-3y=1}{-8x+5y=9}

$$\left. \begin{array} { l } { 8 x - 3 y = 1 } \\ { - 8 x + 5 y = 9 } \end{array} \right.$$

Whakaoti mō x, y

$$x=2$$

$$y=5$$

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.

$$8x-3y=1,-8x+5y=9$$

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

$$8x-3y=1$$

Me tāpiri 3y ki ngā taha e rua o te whārite.

$$8x=3y+1$$

Whakawehea ngā taha e rua ki te 8.

$$x=\frac{1}{8}\left(3y+1\right)$$

Whakareatia \frac{1}{8} ki te 3y+1.

$$x=\frac{3}{8}y+\frac{1}{8}$$

Whakakapia te \frac{3y+1}{8} mō te x ki tērā atu whārite, -8x+5y=9.

$$-8\left(\frac{3}{8}y+\frac{1}{8}\right)+5y=9$$

Whakareatia -8 ki te \frac{3y+1}{8}.

$$-3y-1+5y=9$$

Tāpiri -3y ki te 5y.

$$2y-1=9$$

Me tāpiri 1 ki ngā taha e rua o te whārite.

$$2y=10$$

Whakawehea ngā taha e rua ki te 2.

$$y=5$$

Whakaurua te 5 mō y ki x=\frac{3}{8}y+\frac{1}{8}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{3}{8}\times 5+\frac{1}{8}$$

Whakareatia \frac{3}{8} ki te 5.

$$x=\frac{15+1}{8}$$

Tāpiri \frac{1}{8} ki te \frac{15}{8} 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.

$$x=2,y=5$$

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.

$$8x-3y=1,-8x+5y=9$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}8&-3\\-8&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1\\9\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}8&-3\\-8&5\end{matrix}\right).

$$inverse(\left(\begin{matrix}8&-3\\-8&5\end{matrix}\right))\left(\begin{matrix}8&-3\\-8&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&-3\\-8&5\end{matrix}\right))\left(\begin{matrix}1\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&-3\\-8&5\end{matrix}\right))\left(\begin{matrix}1\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&-3\\-8&5\end{matrix}\right))\left(\begin{matrix}1\\9\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{8\times 5-\left(-3\left(-8\right)\right)}&-\frac{-3}{8\times 5-\left(-3\left(-8\right)\right)}\\-\frac{-8}{8\times 5-\left(-3\left(-8\right)\right)}&\frac{8}{8\times 5-\left(-3\left(-8\right)\right)}\end{matrix}\right)\left(\begin{matrix}1\\9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{16}&\frac{3}{16}\\\frac{1}{2}&\frac{1}{2}\end{matrix}\right)\left(\begin{matrix}1\\9\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{16}+\frac{3}{16}\times 9\\\frac{1}{2}+\frac{1}{2}\times 9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2\\5\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=2,y=5$$

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.

$$8x-3y=1,-8x+5y=9$$

Kia ōrite ai a 8x me -8x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -8 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 8.

$$-8\times 8x-8\left(-3\right)y=-8,8\left(-8\right)x+8\times 5y=8\times 9$$

Whakarūnātia.

$$-64x+24y=-8,-64x+40y=72$$

Me tango -64x+40y=72 mai i -64x+24y=-8 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-64x+64x+24y-40y=-8-72$$

Tāpiri -64x ki te 64x. Ka whakakore atu ngā kupu -64x me 64x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$24y-40y=-8-72$$

Tāpiri 24y ki te -40y.

$$-16y=-8-72$$

Tāpiri -8 ki te -72.

$$-16y=-80$$

Whakawehea ngā taha e rua ki te -16.

$$y=5$$

Whakaurua te 5 mō y ki -8x+5y=9. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-8x+5\times 5=9$$

Whakareatia 5 ki te 5.

$$-8x+25=9$$

Me tango 25 mai i ngā taha e rua o te whārite.

$$-8x=-16$$

Whakawehea ngā taha e rua ki te -8.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=5$$