Whakaoti i te {l}{4x+12y=-24}{-8x-20y=36}

$$\left. \begin{array} { l } { 4 x + 12 y = - 24 } \\ { - 8 x - 20 y = 36 } \end{array} \right.$$

Whakaoti mō x, y

$$x=3$$

$$y=-3$$

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.

$$4x+12y=-24,-8x-20y=36$$

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

$$4x+12y=-24$$

Me tango 12y mai i ngā taha e rua o te whārite.

$$4x=-12y-24$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{1}{4}\left(-12y-24\right)$$

Whakareatia \frac{1}{4} ki te -12y-24.

$$x=-3y-6$$

Whakakapia te -3y-6 mō te x ki tērā atu whārite, -8x-20y=36.

$$-8\left(-3y-6\right)-20y=36$$

Whakareatia -8 ki te -3y-6.

$$24y+48-20y=36$$

Tāpiri 24y ki te -20y.

$$4y+48=36$$

Me tango 48 mai i ngā taha e rua o te whārite.

$$4y=-12$$

Whakawehea ngā taha e rua ki te 4.

$$y=-3$$

Whakaurua te -3 mō y ki x=-3y-6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-3\left(-3\right)-6$$

Whakareatia -3 ki te -3.

$$x=9-6$$

Tāpiri -6 ki te 9.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=-3$$

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.

$$4x+12y=-24,-8x-20y=36$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&12\\-8&-20\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-24\\36\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4&12\\-8&-20\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&12\\-8&-20\end{matrix}\right))\left(\begin{matrix}4&12\\-8&-20\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&12\\-8&-20\end{matrix}\right))\left(\begin{matrix}-24\\36\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&12\\-8&-20\end{matrix}\right))\left(\begin{matrix}-24\\36\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&12\\-8&-20\end{matrix}\right))\left(\begin{matrix}-24\\36\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{20}{4\left(-20\right)-12\left(-8\right)}&-\frac{12}{4\left(-20\right)-12\left(-8\right)}\\-\frac{-8}{4\left(-20\right)-12\left(-8\right)}&\frac{4}{4\left(-20\right)-12\left(-8\right)}\end{matrix}\right)\left(\begin{matrix}-24\\36\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{4}&-\frac{3}{4}\\\frac{1}{2}&\frac{1}{4}\end{matrix}\right)\left(\begin{matrix}-24\\36\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{4}\left(-24\right)-\frac{3}{4}\times 36\\\frac{1}{2}\left(-24\right)+\frac{1}{4}\times 36\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=3,y=-3$$

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.

$$4x+12y=-24,-8x-20y=36$$

Kia ōrite ai a 4x 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 4.

$$-8\times 4x-8\times 12y=-8\left(-24\right),4\left(-8\right)x+4\left(-20\right)y=4\times 36$$

Whakarūnātia.

$$-32x-96y=192,-32x-80y=144$$

Me tango -32x-80y=144 mai i -32x-96y=192 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-32x+32x-96y+80y=192-144$$

Tāpiri -32x ki te 32x. Ka whakakore atu ngā kupu -32x me 32x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-96y+80y=192-144$$

Tāpiri -96y ki te 80y.

$$-16y=192-144$$

Tāpiri 192 ki te -144.

$$-16y=48$$

Whakawehea ngā taha e rua ki te -16.

$$y=-3$$

Whakaurua te -3 mō y ki -8x-20y=36. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-8x-20\left(-3\right)=36$$

Whakareatia -20 ki te -3.

$$-8x+60=36$$

Me tango 60 mai i ngā taha e rua o te whārite.

$$-8x=-24$$

Whakawehea ngā taha e rua ki te -8.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=-3$$