Whakaoti i te {l}{4a-2b+4=0}{64a+8b+4=0}

$$\left. \begin{array} { l } { 4 a - 2 b + 4 = 0 } \\ { 64 a + 8 b + 4 = 0 } \end{array} \right.$$

Whakaoti mō a, b

$$a=-\frac{1}{4}=-0.25$$

$$b = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

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.

$$4a-2b+4=0,64a+8b+4=0$$

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

$$4a-2b+4=0$$

Me tango 4 mai i ngā taha e rua o te whārite.

$$4a-2b=-4$$

Me tāpiri 2b ki ngā taha e rua o te whārite.

$$4a=2b-4$$

Whakawehea ngā taha e rua ki te 4.

$$a=\frac{1}{4}\left(2b-4\right)$$

Whakareatia \frac{1}{4} ki te -4+2b.

$$a=\frac{1}{2}b-1$$

Whakakapia te \frac{b}{2}-1 mō te a ki tērā atu whārite, 64a+8b+4=0.

$$64\left(\frac{1}{2}b-1\right)+8b+4=0$$

Whakareatia 64 ki te \frac{b}{2}-1.

$$32b-64+8b+4=0$$

Tāpiri 32b ki te 8b.

$$40b-64+4=0$$

Tāpiri -64 ki te 4.

$$40b-60=0$$

Me tāpiri 60 ki ngā taha e rua o te whārite.

$$40b=60$$

Whakawehea ngā taha e rua ki te 40.

$$b=\frac{3}{2}$$

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

$$a=\frac{1}{2}\times \frac{3}{2}-1$$

Whakareatia \frac{1}{2} ki te \frac{3}{2} 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.

$$a=\frac{3}{4}-1$$

Tāpiri -1 ki te \frac{3}{4}.

$$a=-\frac{1}{4}$$

Kua oti te pūnaha te whakatau.

$$a=-\frac{1}{4},b=\frac{3}{2}$$

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.

$$4a-2b+4=0,64a+8b+4=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&-2\\64&8\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-4\\-4\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4&-2\\64&8\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&-2\\64&8\end{matrix}\right))\left(\begin{matrix}4&-2\\64&8\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}4&-2\\64&8\end{matrix}\right))\left(\begin{matrix}-4\\-4\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}4&-2\\64&8\end{matrix}\right))\left(\begin{matrix}-4\\-4\end{matrix}\right)$$

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

$$\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}4&-2\\64&8\end{matrix}\right))\left(\begin{matrix}-4\\-4\end{matrix}\right)$$

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.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{8}{4\times 8-\left(-2\times 64\right)}&-\frac{-2}{4\times 8-\left(-2\times 64\right)}\\-\frac{64}{4\times 8-\left(-2\times 64\right)}&\frac{4}{4\times 8-\left(-2\times 64\right)}\end{matrix}\right)\left(\begin{matrix}-4\\-4\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{1}{20}&\frac{1}{80}\\-\frac{2}{5}&\frac{1}{40}\end{matrix}\right)\left(\begin{matrix}-4\\-4\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{1}{20}\left(-4\right)+\frac{1}{80}\left(-4\right)\\-\frac{2}{5}\left(-4\right)+\frac{1}{40}\left(-4\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{4}\\\frac{3}{2}\end{matrix}\right)$$

Tangohia ngā huānga poukapa a me b.

$$a=-\frac{1}{4},b=\frac{3}{2}$$

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.

$$4a-2b+4=0,64a+8b+4=0$$

Kia ōrite ai a 4a me 64a, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 64 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 4.

$$64\times 4a+64\left(-2\right)b+64\times 4=0,4\times 64a+4\times 8b+4\times 4=0$$

Whakarūnātia.

$$256a-128b+256=0,256a+32b+16=0$$

Me tango 256a+32b+16=0 mai i 256a-128b+256=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$256a-256a-128b-32b+256-16=0$$

Tāpiri 256a ki te -256a. Ka whakakore atu ngā kupu 256a me -256a, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-128b-32b+256-16=0$$

Tāpiri -128b ki te -32b.

$$-160b+256-16=0$$

Tāpiri 256 ki te -16.

$$-160b+240=0$$

Me tango 240 mai i ngā taha e rua o te whārite.

$$-160b=-240$$

Whakawehea ngā taha e rua ki te -160.

$$b=\frac{3}{2}$$

Whakaurua te \frac{3}{2} mō b ki 64a+8b+4=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$64a+8\times \frac{3}{2}+4=0$$

Whakareatia 8 ki te \frac{3}{2}.

$$64a+12+4=0$$

Tāpiri 12 ki te 4.

$$64a+16=0$$

Me tango 16 mai i ngā taha e rua o te whārite.

$$64a=-16$$

Whakawehea ngā taha e rua ki te 64.

$$a=-\frac{1}{4}$$

Kua oti te pūnaha te whakatau.

$$a=-\frac{1}{4},b=\frac{3}{2}$$