Whakaoti i te {l}{8x+4y=2}{2x+3y=5}

$$\left. \begin{array} { l } { 8 x + 4 y = 2 } \\ { 2 x + 3 y = 5 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-\frac{7}{8}=-0.875$$

$$y = \frac{9}{4} = 2\frac{1}{4} = 2.25$$

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.

$$8x+4y=2,2x+3y=5$$

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 tango 4y mai i 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=5.

$$2\left(-\frac{1}{2}y+\frac{1}{4}\right)+3y=5$$

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

$$-y+\frac{1}{2}+3y=5$$

Tāpiri -y ki te 3y.

$$2y+\frac{1}{2}=5$$

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

$$2y=\frac{9}{2}$$

Whakawehea ngā taha e rua ki te 2.

$$y=\frac{9}{4}$$

Whakaurua te \frac{9}{4} 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.

$$x=-\frac{1}{2}\times \frac{9}{4}+\frac{1}{4}$$

Whakareatia -\frac{1}{2} ki te \frac{9}{4} 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.

$$x=-\frac{9}{8}+\frac{1}{4}$$

Tāpiri \frac{1}{4} ki te -\frac{9}{8} 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.

$$x=-\frac{7}{8}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{7}{8},y=\frac{9}{4}$$

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.

$$8x+4y=2,2x+3y=5$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&4\\2&3\end{matrix}\right))\left(\begin{matrix}2\\5\end{matrix}\right)$$

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{16}&-\frac{1}{4}\\-\frac{1}{8}&\frac{1}{2}\end{matrix}\right)\left(\begin{matrix}2\\5\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{16}\times 2-\frac{1}{4}\times 5\\-\frac{1}{8}\times 2+\frac{1}{2}\times 5\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{7}{8}\\\frac{9}{4}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{7}{8},y=\frac{9}{4}$$

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.

$$8x+4y=2,2x+3y=5$$

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.

$$2\times 8x+2\times 4y=2\times 2,8\times 2x+8\times 3y=8\times 5$$

Whakarūnātia.

$$16x+8y=4,16x+24y=40$$

Me tango 16x+24y=40 mai i 16x+8y=4 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$16x-16x+8y-24y=4-40$$

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.

$$8y-24y=4-40$$

Tāpiri 8y ki te -24y.

$$-16y=4-40$$

Tāpiri 4 ki te -40.

$$-16y=-36$$

Whakawehea ngā taha e rua ki te -16.

$$y=\frac{9}{4}$$

Whakaurua te \frac{9}{4} mō y ki 2x+3y=5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x+3\times \frac{9}{4}=5$$

Whakareatia 3 ki te \frac{9}{4}.

$$2x+\frac{27}{4}=5$$

Me tango \frac{27}{4} mai i ngā taha e rua o te whārite.

$$2x=-\frac{7}{4}$$

Whakawehea ngā taha e rua ki te 2.

$$x=-\frac{7}{8}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{7}{8},y=\frac{9}{4}$$