Whakaoti i te {l}{5x+7y=71}{8x-3y=10}

$$\left. \begin{array} { l } { 5 x + 7 y = 71 } \\ { 8 x - 3 y = 10 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{283}{71} = 3\frac{70}{71} \approx 3.985915493$$

$$y = \frac{518}{71} = 7\frac{21}{71} \approx 7.295774648$$

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.

$$5x+7y=71,8x-3y=10$$

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

$$5x+7y=71$$

Me tango 7y mai i ngā taha e rua o te whārite.

$$5x=-7y+71$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\left(-7y+71\right)$$

Whakareatia \frac{1}{5} ki te -7y+71.

$$x=-\frac{7}{5}y+\frac{71}{5}$$

Whakakapia te \frac{-7y+71}{5} mō te x ki tērā atu whārite, 8x-3y=10.

$$8\left(-\frac{7}{5}y+\frac{71}{5}\right)-3y=10$$

Whakareatia 8 ki te \frac{-7y+71}{5}.

$$-\frac{56}{5}y+\frac{568}{5}-3y=10$$

Tāpiri -\frac{56y}{5} ki te -3y.

$$-\frac{71}{5}y+\frac{568}{5}=10$$

Me tango \frac{568}{5} mai i ngā taha e rua o te whārite.

$$-\frac{71}{5}y=-\frac{518}{5}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{71}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{518}{71}$$

Whakaurua te \frac{518}{71} mō y ki x=-\frac{7}{5}y+\frac{71}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{7}{5}\times \frac{518}{71}+\frac{71}{5}$$

Whakareatia -\frac{7}{5} ki te \frac{518}{71} 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.

$$x=-\frac{3626}{355}+\frac{71}{5}$$

Tāpiri \frac{71}{5} ki te -\frac{3626}{355} 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.

$$x=\frac{283}{71}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{283}{71},y=\frac{518}{71}$$

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.

$$5x+7y=71,8x-3y=10$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&7\\8&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}71\\10\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&7\\8&-3\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&7\\8&-3\end{matrix}\right))\left(\begin{matrix}5&7\\8&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&7\\8&-3\end{matrix}\right))\left(\begin{matrix}71\\10\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&7\\8&-3\end{matrix}\right))\left(\begin{matrix}71\\10\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&7\\8&-3\end{matrix}\right))\left(\begin{matrix}71\\10\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{5\left(-3\right)-7\times 8}&-\frac{7}{5\left(-3\right)-7\times 8}\\-\frac{8}{5\left(-3\right)-7\times 8}&\frac{5}{5\left(-3\right)-7\times 8}\end{matrix}\right)\left(\begin{matrix}71\\10\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{71}&\frac{7}{71}\\\frac{8}{71}&-\frac{5}{71}\end{matrix}\right)\left(\begin{matrix}71\\10\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{71}\times 71+\frac{7}{71}\times 10\\\frac{8}{71}\times 71-\frac{5}{71}\times 10\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{283}{71}\\\frac{518}{71}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{283}{71},y=\frac{518}{71}$$

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.

$$5x+7y=71,8x-3y=10$$

Kia ōrite ai a 5x me 8x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 8 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 5.

$$8\times 5x+8\times 7y=8\times 71,5\times 8x+5\left(-3\right)y=5\times 10$$

Whakarūnātia.

$$40x+56y=568,40x-15y=50$$

Me tango 40x-15y=50 mai i 40x+56y=568 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$40x-40x+56y+15y=568-50$$

Tāpiri 40x ki te -40x. Ka whakakore atu ngā kupu 40x me -40x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$56y+15y=568-50$$

Tāpiri 56y ki te 15y.

$$71y=568-50$$

Tāpiri 568 ki te -50.

$$71y=518$$

Whakawehea ngā taha e rua ki te 71.

$$y=\frac{518}{71}$$

Whakaurua te \frac{518}{71} mō y ki 8x-3y=10. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$8x-3\times \frac{518}{71}=10$$

Whakareatia -3 ki te \frac{518}{71}.

$$8x-\frac{1554}{71}=10$$

Me tāpiri \frac{1554}{71} ki ngā taha e rua o te whārite.

$$8x=\frac{2264}{71}$$

Whakawehea ngā taha e rua ki te 8.

$$x=\frac{283}{71}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{283}{71},y=\frac{518}{71}$$