Whakaoti i te {l}{3y=4x+8}{2y-8x=7}

$$\left\{ \begin{array} { l } { 3 y = 4 x + 8 } \\ { 2 y - 8 x = 7 } \end{array} \right.$$

Whakaoti mō y, x

$$x=-\frac{5}{16}=-0.3125$$

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

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tangohia te 4x mai i ngā taha e rua.

$$3y-4x=8$$

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.

$$3y-4x=8,2y-8x=7$$

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

$$3y-4x=8$$

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

$$3y=4x+8$$

Whakawehea ngā taha e rua ki te 3.

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

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

$$y=\frac{4}{3}x+\frac{8}{3}$$

Whakakapia te \frac{8+4x}{3} mō te y ki tērā atu whārite, 2y-8x=7.

$$2\left(\frac{4}{3}x+\frac{8}{3}\right)-8x=7$$

Whakareatia 2 ki te \frac{8+4x}{3}.

$$\frac{8}{3}x+\frac{16}{3}-8x=7$$

Tāpiri \frac{8x}{3} ki te -8x.

$$-\frac{16}{3}x+\frac{16}{3}=7$$

Me tango \frac{16}{3} mai i ngā taha e rua o te whārite.

$$-\frac{16}{3}x=\frac{5}{3}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{16}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=-\frac{5}{16}$$

Whakaurua te -\frac{5}{16} mō x ki y=\frac{4}{3}x+\frac{8}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=\frac{4}{3}\left(-\frac{5}{16}\right)+\frac{8}{3}$$

Whakareatia \frac{4}{3} ki te -\frac{5}{16} 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.

$$y=-\frac{5}{12}+\frac{8}{3}$$

Tāpiri \frac{8}{3} ki te -\frac{5}{12} 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.

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

Kua oti te pūnaha te whakatau.

$$y=\frac{9}{4},x=-\frac{5}{16}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tangohia te 4x mai i ngā taha e rua.

$$3y-4x=8$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$3y-4x=8,2y-8x=7$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&-4\\2&-8\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}8\\7\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}3&-4\\2&-8\end{matrix}\right))\left(\begin{matrix}3&-4\\2&-8\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}3&-4\\2&-8\end{matrix}\right))\left(\begin{matrix}8\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}3&-4\\2&-8\end{matrix}\right))\left(\begin{matrix}8\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}3&-4\\2&-8\end{matrix}\right))\left(\begin{matrix}8\\7\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{8}{3\left(-8\right)-\left(-4\times 2\right)}&-\frac{-4}{3\left(-8\right)-\left(-4\times 2\right)}\\-\frac{2}{3\left(-8\right)-\left(-4\times 2\right)}&\frac{3}{3\left(-8\right)-\left(-4\times 2\right)}\end{matrix}\right)\left(\begin{matrix}8\\7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}&-\frac{1}{4}\\\frac{1}{8}&-\frac{3}{16}\end{matrix}\right)\left(\begin{matrix}8\\7\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}\times 8-\frac{1}{4}\times 7\\\frac{1}{8}\times 8-\frac{3}{16}\times 7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{9}{4}\\-\frac{5}{16}\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=\frac{9}{4},x=-\frac{5}{16}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Tangohia te 4x mai i ngā taha e rua.

$$3y-4x=8$$

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.

$$3y-4x=8,2y-8x=7$$

Kia ōrite ai a 3y me 2y, 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 3.

$$2\times 3y+2\left(-4\right)x=2\times 8,3\times 2y+3\left(-8\right)x=3\times 7$$

Whakarūnātia.

$$6y-8x=16,6y-24x=21$$

Me tango 6y-24x=21 mai i 6y-8x=16 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6y-6y-8x+24x=16-21$$

Tāpiri 6y ki te -6y. Ka whakakore atu ngā kupu 6y me -6y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-8x+24x=16-21$$

Tāpiri -8x ki te 24x.

$$16x=16-21$$

Tāpiri 16 ki te -21.

$$16x=-5$$

Whakawehea ngā taha e rua ki te 16.

$$x=-\frac{5}{16}$$

Whakaurua te -\frac{5}{16} mō x ki 2y-8x=7. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$2y-8\left(-\frac{5}{16}\right)=7$$

Whakareatia -8 ki te -\frac{5}{16}.

$$2y+\frac{5}{2}=7$$

Me tango \frac{5}{2} mai i ngā taha e rua o te whārite.

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

Whakawehea ngā taha e rua ki te 2.

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

Kua oti te pūnaha te whakatau.

$$y=\frac{9}{4},x=-\frac{5}{16}$$