Whakaoti i te {l}{8x-3y=4}{-4x+4y=8}

$$\left. \begin{array} { l } { 8 x - 3 y = 4 } \\ { - 4 x + 4 y = 8 } \end{array} \right.$$

Whakaoti mō x, y

$$x=2$$

$$y=4$$

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.

$$8x-3y=4,-4x+4y=8$$

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

$$8x-3y=4$$

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

$$8x=3y+4$$

Whakawehea ngā taha e rua ki te 8.

$$x=\frac{1}{8}\left(3y+4\right)$$

Whakareatia \frac{1}{8} ki te 3y+4.

$$x=\frac{3}{8}y+\frac{1}{2}$$

Whakakapia te \frac{3y}{8}+\frac{1}{2} mō te x ki tērā atu whārite, -4x+4y=8.

$$-4\left(\frac{3}{8}y+\frac{1}{2}\right)+4y=8$$

Whakareatia -4 ki te \frac{3y}{8}+\frac{1}{2}.

$$-\frac{3}{2}y-2+4y=8$$

Tāpiri -\frac{3y}{2} ki te 4y.

$$\frac{5}{2}y-2=8$$

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

$$\frac{5}{2}y=10$$

Whakawehea ngā taha e rua o te whārite ki te \frac{5}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=4$$

Whakaurua te 4 mō y ki x=\frac{3}{8}y+\frac{1}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{3}{8}\times 4+\frac{1}{2}$$

Whakareatia \frac{3}{8} ki te 4.

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

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

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=4$$

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.

$$8x-3y=4,-4x+4y=8$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}8&-3\\-4&4\end{matrix}\right).

$$inverse(\left(\begin{matrix}8&-3\\-4&4\end{matrix}\right))\left(\begin{matrix}8&-3\\-4&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&-3\\-4&4\end{matrix}\right))\left(\begin{matrix}4\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&-3\\-4&4\end{matrix}\right))\left(\begin{matrix}4\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&-3\\-4&4\end{matrix}\right))\left(\begin{matrix}4\\8\end{matrix}\right)$$

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{5}&\frac{3}{20}\\\frac{1}{5}&\frac{2}{5}\end{matrix}\right)\left(\begin{matrix}4\\8\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{5}\times 4+\frac{3}{20}\times 8\\\frac{1}{5}\times 4+\frac{2}{5}\times 8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2\\4\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=2,y=4$$

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.

$$8x-3y=4,-4x+4y=8$$

Kia ōrite ai a 8x me -4x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -4 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 8.

$$-4\times 8x-4\left(-3\right)y=-4\times 4,8\left(-4\right)x+8\times 4y=8\times 8$$

Whakarūnātia.

$$-32x+12y=-16,-32x+32y=64$$

Me tango -32x+32y=64 mai i -32x+12y=-16 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-32x+32x+12y-32y=-16-64$$

Tāpiri -32x ki te 32x. Ka whakakore atu ngā kupu -32x me 32x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$12y-32y=-16-64$$

Tāpiri 12y ki te -32y.

$$-20y=-16-64$$

Tāpiri -16 ki te -64.

$$-20y=-80$$

Whakawehea ngā taha e rua ki te -20.

$$y=4$$

Whakaurua te 4 mō y ki -4x+4y=8. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-4x+4\times 4=8$$

Whakareatia 4 ki te 4.

$$-4x+16=8$$

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

$$-4x=-8$$

Whakawehea ngā taha e rua ki te -4.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=4$$