Whakaoti i te {l}{7x+5y=12}{8x-2y=7}

$$\left. \begin{array} { l } { 7 x + 5 y = 12 } \\ { 8 x - 2 y = 7 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{59}{54} = 1\frac{5}{54} \approx 1.092592593$$

$$y=\frac{47}{54}\approx 0.87037037$$

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.

$$7x+5y=12,8x-2y=7$$

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

$$7x+5y=12$$

Me tango 5y mai i ngā taha e rua o te whārite.

$$7x=-5y+12$$

Whakawehea ngā taha e rua ki te 7.

$$x=\frac{1}{7}\left(-5y+12\right)$$

Whakareatia \frac{1}{7} ki te -5y+12.

$$x=-\frac{5}{7}y+\frac{12}{7}$$

Whakakapia te \frac{-5y+12}{7} mō te x ki tērā atu whārite, 8x-2y=7.

$$8\left(-\frac{5}{7}y+\frac{12}{7}\right)-2y=7$$

Whakareatia 8 ki te \frac{-5y+12}{7}.

$$-\frac{40}{7}y+\frac{96}{7}-2y=7$$

Tāpiri -\frac{40y}{7} ki te -2y.

$$-\frac{54}{7}y+\frac{96}{7}=7$$

Me tango \frac{96}{7} mai i ngā taha e rua o te whārite.

$$-\frac{54}{7}y=-\frac{47}{7}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{54}{7}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{47}{54}$$

Whakaurua te \frac{47}{54} mō y ki x=-\frac{5}{7}y+\frac{12}{7}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{5}{7}\times \frac{47}{54}+\frac{12}{7}$$

Whakareatia -\frac{5}{7} ki te \frac{47}{54} 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.

$$x=-\frac{235}{378}+\frac{12}{7}$$

Tāpiri \frac{12}{7} ki te -\frac{235}{378} 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.

$$x=\frac{59}{54}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{59}{54},y=\frac{47}{54}$$

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.

$$7x+5y=12,8x-2y=7$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}7&5\\8&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}12\\7\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}7&5\\8&-2\end{matrix}\right).

$$inverse(\left(\begin{matrix}7&5\\8&-2\end{matrix}\right))\left(\begin{matrix}7&5\\8&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&5\\8&-2\end{matrix}\right))\left(\begin{matrix}12\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&5\\8&-2\end{matrix}\right))\left(\begin{matrix}12\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&5\\8&-2\end{matrix}\right))\left(\begin{matrix}12\\7\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{7\left(-2\right)-5\times 8}&-\frac{5}{7\left(-2\right)-5\times 8}\\-\frac{8}{7\left(-2\right)-5\times 8}&\frac{7}{7\left(-2\right)-5\times 8}\end{matrix}\right)\left(\begin{matrix}12\\7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{27}&\frac{5}{54}\\\frac{4}{27}&-\frac{7}{54}\end{matrix}\right)\left(\begin{matrix}12\\7\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{27}\times 12+\frac{5}{54}\times 7\\\frac{4}{27}\times 12-\frac{7}{54}\times 7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{59}{54}\\\frac{47}{54}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{59}{54},y=\frac{47}{54}$$

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.

$$7x+5y=12,8x-2y=7$$

Kia ōrite ai a 7x 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 7.

$$8\times 7x+8\times 5y=8\times 12,7\times 8x+7\left(-2\right)y=7\times 7$$

Whakarūnātia.

$$56x+40y=96,56x-14y=49$$

Me tango 56x-14y=49 mai i 56x+40y=96 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$56x-56x+40y+14y=96-49$$

Tāpiri 56x ki te -56x. Ka whakakore atu ngā kupu 56x me -56x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$40y+14y=96-49$$

Tāpiri 40y ki te 14y.

$$54y=96-49$$

Tāpiri 96 ki te -49.

$$54y=47$$

Whakawehea ngā taha e rua ki te 54.

$$y=\frac{47}{54}$$

Whakaurua te \frac{47}{54} mō y ki 8x-2y=7. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$8x-2\times \frac{47}{54}=7$$

Whakareatia -2 ki te \frac{47}{54}.

$$8x-\frac{47}{27}=7$$

Me tāpiri \frac{47}{27} ki ngā taha e rua o te whārite.

$$8x=\frac{236}{27}$$

Whakawehea ngā taha e rua ki te 8.

$$x=\frac{59}{54}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{59}{54},y=\frac{47}{54}$$