Whakaoti i te {r}{-6x+21y=-24}{+quad6x-4y=24}

$$\left. \begin{array} { r } { - 6 x + 21 y = - 24 } \\ { + \quad 6 x - 4 y = 24 } \end{array} \right.$$

Whakaoti mō x, y

$$x=4$$

$$y=0$$

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.

$$-6x+21y=-24,6x-4y=24$$

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

$$-6x+21y=-24$$

Me tango 21y mai i ngā taha e rua o te whārite.

$$-6x=-21y-24$$

Whakawehea ngā taha e rua ki te -6.

$$x=-\frac{1}{6}\left(-21y-24\right)$$

Whakareatia -\frac{1}{6} ki te -21y-24.

$$x=\frac{7}{2}y+4$$

Whakakapia te \frac{7y}{2}+4 mō te x ki tērā atu whārite, 6x-4y=24.

$$6\left(\frac{7}{2}y+4\right)-4y=24$$

Whakareatia 6 ki te \frac{7y}{2}+4.

$$21y+24-4y=24$$

Tāpiri 21y ki te -4y.

$$17y+24=24$$

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=\frac{7}{2}y+4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=4$$

Kua oti te pūnaha te whakatau.

$$x=4,y=0$$

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.

$$-6x+21y=-24,6x-4y=24$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-6&21\\6&-4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-24\\24\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-6&21\\6&-4\end{matrix}\right).

$$inverse(\left(\begin{matrix}-6&21\\6&-4\end{matrix}\right))\left(\begin{matrix}-6&21\\6&-4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-6&21\\6&-4\end{matrix}\right))\left(\begin{matrix}-24\\24\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-6&21\\6&-4\end{matrix}\right))\left(\begin{matrix}-24\\24\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-6&21\\6&-4\end{matrix}\right))\left(\begin{matrix}-24\\24\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{-6\left(-4\right)-21\times 6}&-\frac{21}{-6\left(-4\right)-21\times 6}\\-\frac{6}{-6\left(-4\right)-21\times 6}&-\frac{6}{-6\left(-4\right)-21\times 6}\end{matrix}\right)\left(\begin{matrix}-24\\24\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{51}&\frac{7}{34}\\\frac{1}{17}&\frac{1}{17}\end{matrix}\right)\left(\begin{matrix}-24\\24\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{51}\left(-24\right)+\frac{7}{34}\times 24\\\frac{1}{17}\left(-24\right)+\frac{1}{17}\times 24\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\\0\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=4,y=0$$

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.

$$-6x+21y=-24,6x-4y=24$$

Kia ōrite ai a -6x me 6x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 6 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te -6.

$$6\left(-6\right)x+6\times 21y=6\left(-24\right),-6\times 6x-6\left(-4\right)y=-6\times 24$$

Whakarūnātia.

$$-36x+126y=-144,-36x+24y=-144$$

Me tango -36x+24y=-144 mai i -36x+126y=-144 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-36x+36x+126y-24y=-144+144$$

Tāpiri -36x ki te 36x. Ka whakakore atu ngā kupu -36x me 36x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$126y-24y=-144+144$$

Tāpiri 126y ki te -24y.

$$102y=-144+144$$

Tāpiri -144 ki te 144.

$$102y=0$$

Whakawehea ngā taha e rua ki te 102.

$$y=0$$

Whakaurua te 0 mō y ki 6x-4y=24. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$6x=24$$

Whakawehea ngā taha e rua ki te 6.

$$x=4$$

Kua oti te pūnaha te whakatau.

$$x=4,y=0$$