Whakaoti i te {l}{-6x+5y=1}{6x+4y=-10}

$$\left. \begin{array} { l } { - 6 x + 5 y = 1 } \\ { 6 x + 4 y = - 10 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-1$$

$$y=-1$$

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.

$$-6x+5y=1,6x+4y=-10$$

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

$$-6x+5y=1$$

Me tango 5y mai i ngā taha e rua o te whārite.

$$-6x=-5y+1$$

Whakawehea ngā taha e rua ki te -6.

$$x=-\frac{1}{6}\left(-5y+1\right)$$

Whakareatia -\frac{1}{6} ki te -5y+1.

$$x=\frac{5}{6}y-\frac{1}{6}$$

Whakakapia te \frac{5y-1}{6} mō te x ki tērā atu whārite, 6x+4y=-10.

$$6\left(\frac{5}{6}y-\frac{1}{6}\right)+4y=-10$$

Whakareatia 6 ki te \frac{5y-1}{6}.

$$5y-1+4y=-10$$

Tāpiri 5y ki te 4y.

$$9y-1=-10$$

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

$$9y=-9$$

Whakawehea ngā taha e rua ki te 9.

$$y=-1$$

Whakaurua te -1 mō y ki x=\frac{5}{6}y-\frac{1}{6}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{5}{6}\left(-1\right)-\frac{1}{6}$$

Whakareatia \frac{5}{6} ki te -1.

$$x=\frac{-5-1}{6}$$

Tāpiri -\frac{1}{6} ki te -\frac{5}{6} 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.

$$x=-1$$

Kua oti te pūnaha te whakatau.

$$x=-1,y=-1$$

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.

$$-6x+5y=1,6x+4y=-10$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-6&5\\6&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1\\-10\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-6&5\\6&4\end{matrix}\right).

$$inverse(\left(\begin{matrix}-6&5\\6&4\end{matrix}\right))\left(\begin{matrix}-6&5\\6&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-6&5\\6&4\end{matrix}\right))\left(\begin{matrix}1\\-10\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-6&5\\6&4\end{matrix}\right))\left(\begin{matrix}1\\-10\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-6&5\\6&4\end{matrix}\right))\left(\begin{matrix}1\\-10\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{-6\times 4-5\times 6}&-\frac{5}{-6\times 4-5\times 6}\\-\frac{6}{-6\times 4-5\times 6}&-\frac{6}{-6\times 4-5\times 6}\end{matrix}\right)\left(\begin{matrix}1\\-10\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{27}&\frac{5}{54}\\\frac{1}{9}&\frac{1}{9}\end{matrix}\right)\left(\begin{matrix}1\\-10\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{27}+\frac{5}{54}\left(-10\right)\\\frac{1}{9}+\frac{1}{9}\left(-10\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=-1,y=-1$$

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.

$$-6x+5y=1,6x+4y=-10$$

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.

$$6\left(-6\right)x+6\times 5y=6,-6\times 6x-6\times 4y=-6\left(-10\right)$$

Whakarūnātia.

$$-36x+30y=6,-36x-24y=60$$

Me tango -36x-24y=60 mai i -36x+30y=6 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-36x+36x+30y+24y=6-60$$

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.

$$30y+24y=6-60$$

Tāpiri 30y ki te 24y.

$$54y=6-60$$

Tāpiri 6 ki te -60.

$$54y=-54$$

Whakawehea ngā taha e rua ki te 54.

$$y=-1$$

Whakaurua te -1 mō y ki 6x+4y=-10. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$6x+4\left(-1\right)=-10$$

Whakareatia 4 ki te -1.

$$6x-4=-10$$

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

$$6x=-6$$

Whakawehea ngā taha e rua ki te 6.

$$x=-1$$

Kua oti te pūnaha te whakatau.

$$x=-1,y=-1$$