Whakaoti i te {l}{-5y+8x=-18}{5y+2x=58}

$$\left. \begin{array} { l } { - 5 y + 8 x = - 18 } \\ { 5 y + 2 x = 58 } \end{array} \right.$$

Whakaoti mō y, x

$$x=4$$

$$y=10$$

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.

$$-5y+8x=-18,5y+2x=58$$

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

$$-5y+8x=-18$$

Me tango 8x mai i ngā taha e rua o te whārite.

$$-5y=-8x-18$$

Whakawehea ngā taha e rua ki te -5.

$$y=-\frac{1}{5}\left(-8x-18\right)$$

Whakareatia -\frac{1}{5} ki te -8x-18.

$$y=\frac{8}{5}x+\frac{18}{5}$$

Whakakapia te \frac{8x+18}{5} mō te y ki tērā atu whārite, 5y+2x=58.

$$5\left(\frac{8}{5}x+\frac{18}{5}\right)+2x=58$$

Whakareatia 5 ki te \frac{8x+18}{5}.

$$8x+18+2x=58$$

Tāpiri 8x ki te 2x.

$$10x+18=58$$

Me tango 18 mai i ngā taha e rua o te whārite.

$$10x=40$$

Whakawehea ngā taha e rua ki te 10.

$$x=4$$

Whakaurua te 4 mō x ki y=\frac{8}{5}x+\frac{18}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=\frac{8}{5}\times 4+\frac{18}{5}$$

Whakareatia \frac{8}{5} ki te 4.

$$y=\frac{32+18}{5}$$

Tāpiri \frac{18}{5} ki te \frac{32}{5} 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.

$$y=10$$

Kua oti te pūnaha te whakatau.

$$y=10,x=4$$

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.

$$-5y+8x=-18,5y+2x=58$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-5&8\\5&2\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-18\\58\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-5&8\\5&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}-5&8\\5&2\end{matrix}\right))\left(\begin{matrix}-5&8\\5&2\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}-5&8\\5&2\end{matrix}\right))\left(\begin{matrix}-18\\58\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}-5&8\\5&2\end{matrix}\right))\left(\begin{matrix}-18\\58\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}-5&8\\5&2\end{matrix}\right))\left(\begin{matrix}-18\\58\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{2}{-5\times 2-8\times 5}&-\frac{8}{-5\times 2-8\times 5}\\-\frac{5}{-5\times 2-8\times 5}&-\frac{5}{-5\times 2-8\times 5}\end{matrix}\right)\left(\begin{matrix}-18\\58\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{25}&\frac{4}{25}\\\frac{1}{10}&\frac{1}{10}\end{matrix}\right)\left(\begin{matrix}-18\\58\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{25}\left(-18\right)+\frac{4}{25}\times 58\\\frac{1}{10}\left(-18\right)+\frac{1}{10}\times 58\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}10\\4\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=10,x=4$$

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.

$$-5y+8x=-18,5y+2x=58$$

Kia ōrite ai a -5y me 5y, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 5 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te -5.

$$5\left(-5\right)y+5\times 8x=5\left(-18\right),-5\times 5y-5\times 2x=-5\times 58$$

Whakarūnātia.

$$-25y+40x=-90,-25y-10x=-290$$

Me tango -25y-10x=-290 mai i -25y+40x=-90 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-25y+25y+40x+10x=-90+290$$

Tāpiri -25y ki te 25y. Ka whakakore atu ngā kupu -25y me 25y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$40x+10x=-90+290$$

Tāpiri 40x ki te 10x.

$$50x=-90+290$$

Tāpiri -90 ki te 290.

$$50x=200$$

Whakawehea ngā taha e rua ki te 50.

$$x=4$$

Whakaurua te 4 mō x ki 5y+2x=58. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$5y+2\times 4=58$$

Whakareatia 2 ki te 4.

$$5y+8=58$$

Me tango 8 mai i ngā taha e rua o te whārite.

$$5y=50$$

Whakawehea ngā taha e rua ki te 5.

$$y=10$$

Kua oti te pūnaha te whakatau.

$$y=10,x=4$$