Whakaoti i te 6x-3y=10,-3x=6y+5

$$6 x - 3 y = 10 , - 3 x = 6 y + 5$$

Whakaoti mō x, y

$$x=1$$

$$y = -\frac{4}{3} = -1\frac{1}{3} \approx -1.333333333$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Tangohia te 6y mai i ngā taha e rua.

$$-3x-6y=5$$

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.

$$6x-3y=10,-3x-6y=5$$

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

$$6x-3y=10$$

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

$$6x=3y+10$$

Whakawehea ngā taha e rua ki te 6.

$$x=\frac{1}{6}\left(3y+10\right)$$

Whakareatia \frac{1}{6} ki te 3y+10.

$$x=\frac{1}{2}y+\frac{5}{3}$$

Whakakapia te \frac{y}{2}+\frac{5}{3} mō te x ki tērā atu whārite, -3x-6y=5.

$$-3\left(\frac{1}{2}y+\frac{5}{3}\right)-6y=5$$

Whakareatia -3 ki te \frac{y}{2}+\frac{5}{3}.

$$-\frac{3}{2}y-5-6y=5$$

Tāpiri -\frac{3y}{2} ki te -6y.

$$-\frac{15}{2}y-5=5$$

Me tāpiri 5 ki ngā taha e rua o te whārite.

$$-\frac{15}{2}y=10$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{15}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-\frac{4}{3}$$

Whakaurua te -\frac{4}{3} mō y ki x=\frac{1}{2}y+\frac{5}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{1}{2}\left(-\frac{4}{3}\right)+\frac{5}{3}$$

Whakareatia \frac{1}{2} ki te -\frac{4}{3} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=\frac{-2+5}{3}$$

Tāpiri \frac{5}{3} ki te -\frac{2}{3} 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.

$$x=1,y=-\frac{4}{3}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Tangohia te 6y mai i ngā taha e rua.

$$-3x-6y=5$$

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

$$6x-3y=10,-3x-6y=5$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}6&-3\\-3&-6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}10\\5\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}6&-3\\-3&-6\end{matrix}\right))\left(\begin{matrix}6&-3\\-3&-6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&-3\\-3&-6\end{matrix}\right))\left(\begin{matrix}10\\5\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&-3\\-3&-6\end{matrix}\right))\left(\begin{matrix}10\\5\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&-3\\-3&-6\end{matrix}\right))\left(\begin{matrix}10\\5\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{6}{6\left(-6\right)-\left(-3\left(-3\right)\right)}&-\frac{-3}{6\left(-6\right)-\left(-3\left(-3\right)\right)}\\-\frac{-3}{6\left(-6\right)-\left(-3\left(-3\right)\right)}&\frac{6}{6\left(-6\right)-\left(-3\left(-3\right)\right)}\end{matrix}\right)\left(\begin{matrix}10\\5\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{15}&-\frac{1}{15}\\-\frac{1}{15}&-\frac{2}{15}\end{matrix}\right)\left(\begin{matrix}10\\5\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{15}\times 10-\frac{1}{15}\times 5\\-\frac{1}{15}\times 10-\frac{2}{15}\times 5\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1\\-\frac{4}{3}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=1,y=-\frac{4}{3}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Tangohia te 6y mai i ngā taha e rua.

$$-3x-6y=5$$

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.

$$6x-3y=10,-3x-6y=5$$

Kia ōrite ai a 6x me -3x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -3 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 6.

$$-3\times 6x-3\left(-3\right)y=-3\times 10,6\left(-3\right)x+6\left(-6\right)y=6\times 5$$

Whakarūnātia.

$$-18x+9y=-30,-18x-36y=30$$

Me tango -18x-36y=30 mai i -18x+9y=-30 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-18x+18x+9y+36y=-30-30$$

Tāpiri -18x ki te 18x. Ka whakakore atu ngā kupu -18x me 18x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$9y+36y=-30-30$$

Tāpiri 9y ki te 36y.

$$45y=-30-30$$

Tāpiri -30 ki te -30.

$$45y=-60$$

Whakawehea ngā taha e rua ki te 45.

$$y=-\frac{4}{3}$$

Whakaurua te -\frac{4}{3} mō y ki -3x-6y=5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-3x-6\left(-\frac{4}{3}\right)=5$$

Whakareatia -6 ki te -\frac{4}{3}.

$$-3x+8=5$$

Me tango 8 mai i ngā taha e rua o te whārite.

$$-3x=-3$$

Whakawehea ngā taha e rua ki te -3.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=-\frac{4}{3}$$