Whakaoti i te {l}{8x+4y=-4}{4x-2y=8}

$$\left. \begin{array} { l } { 8 x + 4 y = - 4 } \\ { 4 x - 2 y = 8 } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{3}{4}=0.75$$

$$y = -\frac{5}{2} = -2\frac{1}{2} = -2.5$$

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.

$$8x+4y=-4,4x-2y=8$$

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

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

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

$$8x=-4y-4$$

Whakawehea ngā taha e rua ki te 8.

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

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

$$x=-\frac{1}{2}y-\frac{1}{2}$$

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

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

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

$$-2y-2-2y=8$$

Tāpiri -2y ki te -2y.

$$-4y-2=8$$

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

$$-4y=10$$

Whakawehea ngā taha e rua ki te -4.

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

Whakaurua te -\frac{5}{2} mō y ki x=-\frac{1}{2}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.

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

Whakareatia -\frac{1}{2} ki te -\frac{5}{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.

$$x=\frac{5}{4}-\frac{1}{2}$$

Tāpiri -\frac{1}{2} ki te \frac{5}{4} 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.

$$x=\frac{3}{4}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{3}{4},y=-\frac{5}{2}$$

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.

$$8x+4y=-4,4x-2y=8$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}8&4\\4&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-4\\8\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}8&4\\4&-2\end{matrix}\right))\left(\begin{matrix}8&4\\4&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&4\\4&-2\end{matrix}\right))\left(\begin{matrix}-4\\8\end{matrix}\right)$$

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

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

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&4\\4&-2\end{matrix}\right))\left(\begin{matrix}-4\\8\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{8\left(-2\right)-4\times 4}&-\frac{4}{8\left(-2\right)-4\times 4}\\-\frac{4}{8\left(-2\right)-4\times 4}&\frac{8}{8\left(-2\right)-4\times 4}\end{matrix}\right)\left(\begin{matrix}-4\\8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{16}&\frac{1}{8}\\\frac{1}{8}&-\frac{1}{4}\end{matrix}\right)\left(\begin{matrix}-4\\8\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{16}\left(-4\right)+\frac{1}{8}\times 8\\\frac{1}{8}\left(-4\right)-\frac{1}{4}\times 8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{4}\\-\frac{5}{2}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{3}{4},y=-\frac{5}{2}$$

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.

$$8x+4y=-4,4x-2y=8$$

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.

$$4\times 8x+4\times 4y=4\left(-4\right),8\times 4x+8\left(-2\right)y=8\times 8$$

Whakarūnātia.

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

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

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

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.

$$16y+16y=-16-64$$

Tāpiri 16y ki te 16y.

$$32y=-16-64$$

Tāpiri -16 ki te -64.

$$32y=-80$$

Whakawehea ngā taha e rua ki te 32.

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

Whakaurua te -\frac{5}{2} mō y ki 4x-2y=8. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x-2\left(-\frac{5}{2}\right)=8$$

Whakareatia -2 ki te -\frac{5}{2}.

$$4x+5=8$$

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

$$4x=3$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{3}{4}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{3}{4},y=-\frac{5}{2}$$