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

$$x = -\frac{38}{3} = -12\frac{2}{3} \approx -12.666666667$$

$$y = \frac{50}{3} = 16\frac{2}{3} \approx 16.666666667$$

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.

$$-6x-18=58$$

Me tāpiri 18 ki ngā taha e rua o te whārite.

$$-6x=76$$

Whakawehea ngā taha e rua ki te -6.

$$x=-\frac{38}{3}$$

Whakaurua te -\frac{38}{3} 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.

$$y=-\frac{8}{5}\left(-\frac{38}{3}\right)-\frac{18}{5}$$

Whakareatia -\frac{8}{5} ki te -\frac{38}{3} 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.

$$y=\frac{304}{15}-\frac{18}{5}$$

Tāpiri -\frac{18}{5} ki te \frac{304}{15} 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.

$$y=\frac{50}{3}$$

Kua oti te pūnaha te whakatau.

$$y=\frac{50}{3},x=-\frac{38}{3}$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{15}&\frac{4}{15}\\\frac{1}{6}&-\frac{1}{6}\end{matrix}\right)\left(\begin{matrix}-18\\58\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{15}\left(-18\right)+\frac{4}{15}\times 58\\\frac{1}{6}\left(-18\right)-\frac{1}{6}\times 58\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{50}{3}\\-\frac{38}{3}\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=\frac{50}{3},x=-\frac{38}{3}$$

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.

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

Me tango 5y+2x=58 mai i 5y+8x=-18 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

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

Tāpiri 5y ki te -5y. Ka whakakore atu ngā kupu 5y me -5y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$8x-2x=-18-58$$

Tāpiri 8x ki te -2x.

$$6x=-18-58$$

Tāpiri -18 ki te -58.

$$6x=-76$$

Whakawehea ngā taha e rua ki te 6.

$$x=-\frac{38}{3}$$

Whakaurua te -\frac{38}{3} 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.

$$5y+2\left(-\frac{38}{3}\right)=58$$

Whakareatia 2 ki te -\frac{38}{3}.

$$5y-\frac{76}{3}=58$$

Me tāpiri \frac{76}{3} ki ngā taha e rua o te whārite.

$$5y=\frac{250}{3}$$

Whakawehea ngā taha e rua ki te 5.

$$y=\frac{50}{3}$$

Kua oti te pūnaha te whakatau.

$$y=\frac{50}{3},x=-\frac{38}{3}$$