Whakaoti i te {l}{8x-4y=2}{2x+3y=6}

$$\left\{ \begin{array} { l } { 8 x - 4 y = 2 } \\ { 2 x + 3 y = 6 } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{15}{16}=0.9375$$

$$y = \frac{11}{8} = 1\frac{3}{8} = 1.375$$

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.

$$8x-4y=2,2x+3y=6$$

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

$$8x-4y=2$$

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

$$8x=4y+2$$

Whakawehea ngā taha e rua ki te 8.

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

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

$$x=\frac{1}{2}y+\frac{1}{4}$$

Whakakapia te \frac{y}{2}+\frac{1}{4} mō te x ki tērā atu whārite, 2x+3y=6.

$$2\left(\frac{1}{2}y+\frac{1}{4}\right)+3y=6$$

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

$$y+\frac{1}{2}+3y=6$$

Tāpiri y ki te 3y.

$$4y+\frac{1}{2}=6$$

Me tango \frac{1}{2} mai i ngā taha e rua o te whārite.

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

Whakawehea ngā taha e rua ki te 4.

$$y=\frac{11}{8}$$

Whakaurua te \frac{11}{8} mō y ki x=\frac{1}{2}y+\frac{1}{4}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{1}{2}\times \frac{11}{8}+\frac{1}{4}$$

Whakareatia \frac{1}{2} ki te \frac{11}{8} 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.

$$x=\frac{11}{16}+\frac{1}{4}$$

Tāpiri \frac{1}{4} ki te \frac{11}{16} 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.

$$x=\frac{15}{16}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{15}{16},y=\frac{11}{8}$$

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.

$$8x-4y=2,2x+3y=6$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}8&-4\\2&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2\\6\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}8&-4\\2&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}8&-4\\2&3\end{matrix}\right))\left(\begin{matrix}8&-4\\2&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&-4\\2&3\end{matrix}\right))\left(\begin{matrix}2\\6\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&-4\\2&3\end{matrix}\right))\left(\begin{matrix}2\\6\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&-4\\2&3\end{matrix}\right))\left(\begin{matrix}2\\6\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{8\times 3-\left(-4\times 2\right)}&-\frac{-4}{8\times 3-\left(-4\times 2\right)}\\-\frac{2}{8\times 3-\left(-4\times 2\right)}&\frac{8}{8\times 3-\left(-4\times 2\right)}\end{matrix}\right)\left(\begin{matrix}2\\6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{32}&\frac{1}{8}\\-\frac{1}{16}&\frac{1}{4}\end{matrix}\right)\left(\begin{matrix}2\\6\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{32}\times 2+\frac{1}{8}\times 6\\-\frac{1}{16}\times 2+\frac{1}{4}\times 6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{15}{16}\\\frac{11}{8}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{15}{16},y=\frac{11}{8}$$

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.

$$8x-4y=2,2x+3y=6$$

Kia ōrite ai a 8x me 2x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 8.

$$2\times 8x+2\left(-4\right)y=2\times 2,8\times 2x+8\times 3y=8\times 6$$

Whakarūnātia.

$$16x-8y=4,16x+24y=48$$

Me tango 16x+24y=48 mai i 16x-8y=4 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$16x-16x-8y-24y=4-48$$

Tāpiri 16x ki te -16x. Ka whakakore atu ngā kupu 16x me -16x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-8y-24y=4-48$$

Tāpiri -8y ki te -24y.

$$-32y=4-48$$

Tāpiri 4 ki te -48.

$$-32y=-44$$

Whakawehea ngā taha e rua ki te -32.

$$y=\frac{11}{8}$$

Whakaurua te \frac{11}{8} mō y ki 2x+3y=6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x+3\times \frac{11}{8}=6$$

Whakareatia 3 ki te \frac{11}{8}.

$$2x+\frac{33}{8}=6$$

Me tango \frac{33}{8} mai i ngā taha e rua o te whārite.

$$2x=\frac{15}{8}$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{15}{16}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{15}{16},y=\frac{11}{8}$$