Whakaoti i te {l}{2x+5y=17}{6y=6-5x}

$$\left. \begin{array} { l } { 2 x + 5 y = 17 } \\ { 6 y = 6 - 5 x } \end{array} \right.$$

Whakaoti mō x, y

$$x = -\frac{72}{13} = -5\frac{7}{13} \approx -5.538461538$$

$$y = \frac{73}{13} = 5\frac{8}{13} \approx 5.615384615$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Me tāpiri te 5x ki ngā taha e rua.

$$6y+5x=6$$

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.

$$2x+5y=17,5x+6y=6$$

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

$$2x+5y=17$$

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

$$2x=-5y+17$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(-5y+17\right)$$

Whakareatia \frac{1}{2} ki te -5y+17.

$$x=-\frac{5}{2}y+\frac{17}{2}$$

Whakakapia te \frac{-5y+17}{2} mō te x ki tērā atu whārite, 5x+6y=6.

$$5\left(-\frac{5}{2}y+\frac{17}{2}\right)+6y=6$$

Whakareatia 5 ki te \frac{-5y+17}{2}.

$$-\frac{25}{2}y+\frac{85}{2}+6y=6$$

Tāpiri -\frac{25y}{2} ki te 6y.

$$-\frac{13}{2}y+\frac{85}{2}=6$$

Me tango \frac{85}{2} mai i ngā taha e rua o te whārite.

$$-\frac{13}{2}y=-\frac{73}{2}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{13}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{73}{13}$$

Whakaurua te \frac{73}{13} mō y ki x=-\frac{5}{2}y+\frac{17}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{5}{2}\times \frac{73}{13}+\frac{17}{2}$$

Whakareatia -\frac{5}{2} ki te \frac{73}{13} 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.

$$x=-\frac{365}{26}+\frac{17}{2}$$

Tāpiri \frac{17}{2} ki te -\frac{365}{26} 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.

$$x=-\frac{72}{13}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{72}{13},y=\frac{73}{13}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Me tāpiri te 5x ki ngā taha e rua.

$$6y+5x=6$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$2x+5y=17,5x+6y=6$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&5\\5&6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}17\\6\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&5\\5&6\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&5\\5&6\end{matrix}\right))\left(\begin{matrix}2&5\\5&6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&5\\5&6\end{matrix}\right))\left(\begin{matrix}17\\6\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&5\\5&6\end{matrix}\right))\left(\begin{matrix}17\\6\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&5\\5&6\end{matrix}\right))\left(\begin{matrix}17\\6\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{6}{2\times 6-5\times 5}&-\frac{5}{2\times 6-5\times 5}\\-\frac{5}{2\times 6-5\times 5}&\frac{2}{2\times 6-5\times 5}\end{matrix}\right)\left(\begin{matrix}17\\6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{6}{13}&\frac{5}{13}\\\frac{5}{13}&-\frac{2}{13}\end{matrix}\right)\left(\begin{matrix}17\\6\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{6}{13}\times 17+\frac{5}{13}\times 6\\\frac{5}{13}\times 17-\frac{2}{13}\times 6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{72}{13}\\\frac{73}{13}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{72}{13},y=\frac{73}{13}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Me tāpiri te 5x ki ngā taha e rua.

$$6y+5x=6$$

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.

$$2x+5y=17,5x+6y=6$$

Kia ōrite ai a 2x me 5x, 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 2.

$$5\times 2x+5\times 5y=5\times 17,2\times 5x+2\times 6y=2\times 6$$

Whakarūnātia.

$$10x+25y=85,10x+12y=12$$

Me tango 10x+12y=12 mai i 10x+25y=85 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$10x-10x+25y-12y=85-12$$

Tāpiri 10x ki te -10x. Ka whakakore atu ngā kupu 10x me -10x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$25y-12y=85-12$$

Tāpiri 25y ki te -12y.

$$13y=85-12$$

Tāpiri 85 ki te -12.

$$13y=73$$

Whakawehea ngā taha e rua ki te 13.

$$y=\frac{73}{13}$$

Whakaurua te \frac{73}{13} mō y ki 5x+6y=6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$5x+6\times \frac{73}{13}=6$$

Whakareatia 6 ki te \frac{73}{13}.

$$5x+\frac{438}{13}=6$$

Me tango \frac{438}{13} mai i ngā taha e rua o te whārite.

$$5x=-\frac{360}{13}$$

Whakawehea ngā taha e rua ki te 5.

$$x=-\frac{72}{13}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{72}{13},y=\frac{73}{13}$$