Whakaoti i te {l}{6y+4x=27}{x+y=50}

$$\left. \begin{array} { l } { 6 y + 4 x = 27 } \\ { x + y = 50 } \end{array} \right.$$

Whakaoti mō y, x

$$x = \frac{273}{2} = 136\frac{1}{2} = 136.5$$

$$y = -\frac{173}{2} = -86\frac{1}{2} = -86.5$$

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.

$$6y+4x=27,y+x=50$$

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

$$6y+4x=27$$

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

$$6y=-4x+27$$

Whakawehea ngā taha e rua ki te 6.

$$y=\frac{1}{6}\left(-4x+27\right)$$

Whakareatia \frac{1}{6} ki te -4x+27.

$$y=-\frac{2}{3}x+\frac{9}{2}$$

Whakakapia te -\frac{2x}{3}+\frac{9}{2} mō te y ki tērā atu whārite, y+x=50.

$$-\frac{2}{3}x+\frac{9}{2}+x=50$$

Tāpiri -\frac{2x}{3} ki te x.

$$\frac{1}{3}x+\frac{9}{2}=50$$

Me tango \frac{9}{2} mai i ngā taha e rua o te whārite.

$$\frac{1}{3}x=\frac{91}{2}$$

Me whakarea ngā taha e rua ki te 3.

$$x=\frac{273}{2}$$

Whakaurua te \frac{273}{2} mō x ki y=-\frac{2}{3}x+\frac{9}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=-\frac{2}{3}\times \frac{273}{2}+\frac{9}{2}$$

Whakareatia -\frac{2}{3} ki te \frac{273}{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.

$$y=-91+\frac{9}{2}$$

Tāpiri \frac{9}{2} ki te -91.

$$y=-\frac{173}{2}$$

Kua oti te pūnaha te whakatau.

$$y=-\frac{173}{2},x=\frac{273}{2}$$

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.

$$6y+4x=27,y+x=50$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}6&4\\1&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}27\\50\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}6&4\\1&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}6&4\\1&1\end{matrix}\right))\left(\begin{matrix}6&4\\1&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}6&4\\1&1\end{matrix}\right))\left(\begin{matrix}27\\50\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}6&4\\1&1\end{matrix}\right))\left(\begin{matrix}27\\50\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}6&4\\1&1\end{matrix}\right))\left(\begin{matrix}27\\50\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{6-4}&-\frac{4}{6-4}\\-\frac{1}{6-4}&\frac{6}{6-4}\end{matrix}\right)\left(\begin{matrix}27\\50\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}&-2\\-\frac{1}{2}&3\end{matrix}\right)\left(\begin{matrix}27\\50\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}\times 27-2\times 50\\-\frac{1}{2}\times 27+3\times 50\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{173}{2}\\\frac{273}{2}\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=-\frac{173}{2},x=\frac{273}{2}$$

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.

$$6y+4x=27,y+x=50$$

Kia ōrite ai a 6y me y, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 1 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 6.

$$6y+4x=27,6y+6x=6\times 50$$

Whakarūnātia.

$$6y+4x=27,6y+6x=300$$

Me tango 6y+6x=300 mai i 6y+4x=27 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6y-6y+4x-6x=27-300$$

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.

$$4x-6x=27-300$$

Tāpiri 4x ki te -6x.

$$-2x=27-300$$

Tāpiri 27 ki te -300.

$$-2x=-273$$

Whakawehea ngā taha e rua ki te -2.

$$x=\frac{273}{2}$$

Whakaurua te \frac{273}{2} mō x ki y+x=50. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y+\frac{273}{2}=50$$

Me tango \frac{273}{2} mai i ngā taha e rua o te whārite.

$$y=-\frac{173}{2}$$

Kua oti te pūnaha te whakatau.

$$y=-\frac{173}{2},x=\frac{273}{2}$$