Whakaoti i te {l}{3x-6y=9}{y=-8x+24}

$$\left. \begin{array} { l } { 3 x - 6 y = 9 } \\ { y = - 8 x + 24 } \end{array} \right.$$

Whakaoti mō x, y

$$x=3$$

$$y=0$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Me tāpiri te 8x ki ngā taha e rua.

$$y+8x=24$$

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.

$$3x-6y=9,8x+y=24$$

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

$$3x-6y=9$$

Me tāpiri 6y ki ngā taha e rua o te whārite.

$$3x=6y+9$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(6y+9\right)$$

Whakareatia \frac{1}{3} ki te 6y+9.

$$x=2y+3$$

Whakakapia te 2y+3 mō te x ki tērā atu whārite, 8x+y=24.

$$8\left(2y+3\right)+y=24$$

Whakareatia 8 ki te 2y+3.

$$16y+24+y=24$$

Tāpiri 16y ki te y.

$$17y+24=24$$

Me tango 24 mai i ngā taha e rua o te whārite.

$$17y=0$$

Whakawehea ngā taha e rua ki te 17.

$$y=0$$

Whakaurua te 0 mō y ki x=2y+3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=0$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Me tāpiri te 8x ki ngā taha e rua.

$$y+8x=24$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$3x-6y=9,8x+y=24$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&-6\\8&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}9\\24\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&-6\\8&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&-6\\8&1\end{matrix}\right))\left(\begin{matrix}3&-6\\8&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-6\\8&1\end{matrix}\right))\left(\begin{matrix}9\\24\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-6\\8&1\end{matrix}\right))\left(\begin{matrix}9\\24\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-6\\8&1\end{matrix}\right))\left(\begin{matrix}9\\24\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3-\left(-6\times 8\right)}&-\frac{-6}{3-\left(-6\times 8\right)}\\-\frac{8}{3-\left(-6\times 8\right)}&\frac{3}{3-\left(-6\times 8\right)}\end{matrix}\right)\left(\begin{matrix}9\\24\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{51}&\frac{2}{17}\\-\frac{8}{51}&\frac{1}{17}\end{matrix}\right)\left(\begin{matrix}9\\24\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{51}\times 9+\frac{2}{17}\times 24\\-\frac{8}{51}\times 9+\frac{1}{17}\times 24\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3\\0\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=3,y=0$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Me tāpiri te 8x ki ngā taha e rua.

$$y+8x=24$$

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.

$$3x-6y=9,8x+y=24$$

Kia ōrite ai a 3x 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 3.

$$8\times 3x+8\left(-6\right)y=8\times 9,3\times 8x+3y=3\times 24$$

Whakarūnātia.

$$24x-48y=72,24x+3y=72$$

Me tango 24x+3y=72 mai i 24x-48y=72 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$24x-24x-48y-3y=72-72$$

Tāpiri 24x ki te -24x. Ka whakakore atu ngā kupu 24x me -24x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-48y-3y=72-72$$

Tāpiri -48y ki te -3y.

$$-51y=72-72$$

Tāpiri 72 ki te -72.

$$-51y=0$$

Whakawehea ngā taha e rua ki te -51.

$$y=0$$

Whakaurua te 0 mō y ki 8x+y=24. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$8x=24$$

Whakawehea ngā taha e rua ki te 8.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=0$$