Whakaoti i te {l}{2x-4y=10}{6x-4y=11}

$$\left. \begin{array} { l } { 2 x - 4 y = 10 } \\ { 6 x - 4 y = 11 } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{1}{4}=0.25$$

$$y = -\frac{19}{8} = -2\frac{3}{8} = -2.375$$

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.

$$2x-4y=10,6x-4y=11$$

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

$$2x-4y=10$$

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

$$2x=4y+10$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(4y+10\right)$$

Whakareatia \frac{1}{2} ki te 4y+10.

$$x=2y+5$$

Whakakapia te 2y+5 mō te x ki tērā atu whārite, 6x-4y=11.

$$6\left(2y+5\right)-4y=11$$

Whakareatia 6 ki te 2y+5.

$$12y+30-4y=11$$

Tāpiri 12y ki te -4y.

$$8y+30=11$$

Me tango 30 mai i ngā taha e rua o te whārite.

$$8y=-19$$

Whakawehea ngā taha e rua ki te 8.

$$y=-\frac{19}{8}$$

Whakaurua te -\frac{19}{8} mō y ki x=2y+5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=2\left(-\frac{19}{8}\right)+5$$

Whakareatia 2 ki te -\frac{19}{8}.

$$x=-\frac{19}{4}+5$$

Tāpiri 5 ki te -\frac{19}{4}.

$$x=\frac{1}{4}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{1}{4},y=-\frac{19}{8}$$

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.

$$2x-4y=10,6x-4y=11$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&-4\\6&-4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}10\\11\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&-4\\6&-4\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&-4\\6&-4\end{matrix}\right))\left(\begin{matrix}2&-4\\6&-4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-4\\6&-4\end{matrix}\right))\left(\begin{matrix}10\\11\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-4\\6&-4\end{matrix}\right))\left(\begin{matrix}10\\11\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-4\\6&-4\end{matrix}\right))\left(\begin{matrix}10\\11\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{2\left(-4\right)-\left(-4\times 6\right)}&-\frac{-4}{2\left(-4\right)-\left(-4\times 6\right)}\\-\frac{6}{2\left(-4\right)-\left(-4\times 6\right)}&\frac{2}{2\left(-4\right)-\left(-4\times 6\right)}\end{matrix}\right)\left(\begin{matrix}10\\11\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{4}&\frac{1}{4}\\-\frac{3}{8}&\frac{1}{8}\end{matrix}\right)\left(\begin{matrix}10\\11\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{4}\times 10+\frac{1}{4}\times 11\\-\frac{3}{8}\times 10+\frac{1}{8}\times 11\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}\\-\frac{19}{8}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{1}{4},y=-\frac{19}{8}$$

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.

$$2x-4y=10,6x-4y=11$$

Me tango 6x-4y=11 mai i 2x-4y=10 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2x-6x-4y+4y=10-11$$

Tāpiri -4y ki te 4y. Ka whakakore atu ngā kupu -4y me 4y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$2x-6x=10-11$$

Tāpiri 2x ki te -6x.

$$-4x=10-11$$

Tāpiri 10 ki te -11.

$$-4x=-1$$

Whakawehea ngā taha e rua ki te -4.

$$x=\frac{1}{4}$$

Whakaurua te \frac{1}{4} mō x ki 6x-4y=11. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$6\times \frac{1}{4}-4y=11$$

Whakareatia 6 ki te \frac{1}{4}.

$$\frac{3}{2}-4y=11$$

Me tango \frac{3}{2} mai i ngā taha e rua o te whārite.

$$-4y=\frac{19}{2}$$

Whakawehea ngā taha e rua ki te -4.

$$y=-\frac{19}{8}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{1}{4},y=-\frac{19}{8}$$