Whakaoti i te {c}{2x+7y=77}{x-2y=2}

$$\left. \begin{array} { c } { 2 x + 7 y = 77 } \\ { x - 2 y = 2 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{168}{11} = 15\frac{3}{11} \approx 15.272727273$$

$$y = \frac{73}{11} = 6\frac{7}{11} \approx 6.636363636$$

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.

$$2x+7y=77,x-2y=2$$

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

$$2x+7y=77$$

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

$$2x=-7y+77$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(-7y+77\right)$$

Whakareatia \frac{1}{2} ki te -7y+77.

$$x=-\frac{7}{2}y+\frac{77}{2}$$

Whakakapia te \frac{-7y+77}{2} mō te x ki tērā atu whārite, x-2y=2.

$$-\frac{7}{2}y+\frac{77}{2}-2y=2$$

Tāpiri -\frac{7y}{2} ki te -2y.

$$-\frac{11}{2}y+\frac{77}{2}=2$$

Me tango \frac{77}{2} mai i ngā taha e rua o te whārite.

$$-\frac{11}{2}y=-\frac{73}{2}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{11}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{73}{11}$$

Whakaurua te \frac{73}{11} mō y ki x=-\frac{7}{2}y+\frac{77}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{7}{2}\times \frac{73}{11}+\frac{77}{2}$$

Whakareatia -\frac{7}{2} ki te \frac{73}{11} 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.

$$x=-\frac{511}{22}+\frac{77}{2}$$

Tāpiri \frac{77}{2} ki te -\frac{511}{22} 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.

$$x=\frac{168}{11}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{168}{11},y=\frac{73}{11}$$

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.

$$2x+7y=77,x-2y=2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&7\\1&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}77\\2\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&7\\1&-2\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&7\\1&-2\end{matrix}\right))\left(\begin{matrix}2&7\\1&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&7\\1&-2\end{matrix}\right))\left(\begin{matrix}77\\2\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&7\\1&-2\end{matrix}\right))\left(\begin{matrix}77\\2\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&7\\1&-2\end{matrix}\right))\left(\begin{matrix}77\\2\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{2\left(-2\right)-7}&-\frac{7}{2\left(-2\right)-7}\\-\frac{1}{2\left(-2\right)-7}&\frac{2}{2\left(-2\right)-7}\end{matrix}\right)\left(\begin{matrix}77\\2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{11}&\frac{7}{11}\\\frac{1}{11}&-\frac{2}{11}\end{matrix}\right)\left(\begin{matrix}77\\2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{11}\times 77+\frac{7}{11}\times 2\\\frac{1}{11}\times 77-\frac{2}{11}\times 2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{168}{11}\\\frac{73}{11}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{168}{11},y=\frac{73}{11}$$

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.

$$2x+7y=77,x-2y=2$$

Kia ōrite ai a 2x me x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 1 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

$$2x+7y=77,2x+2\left(-2\right)y=2\times 2$$

Whakarūnātia.

$$2x+7y=77,2x-4y=4$$

Me tango 2x-4y=4 mai i 2x+7y=77 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2x-2x+7y+4y=77-4$$

Tāpiri 2x ki te -2x. Ka whakakore atu ngā kupu 2x me -2x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$7y+4y=77-4$$

Tāpiri 7y ki te 4y.

$$11y=77-4$$

Tāpiri 77 ki te -4.

$$11y=73$$

Whakawehea ngā taha e rua ki te 11.

$$y=\frac{73}{11}$$

Whakaurua te \frac{73}{11} mō y ki x-2y=2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x-2\times \frac{73}{11}=2$$

Whakareatia -2 ki te \frac{73}{11}.

$$x-\frac{146}{11}=2$$

Me tāpiri \frac{146}{11} ki ngā taha e rua o te whārite.

$$x=\frac{168}{11}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{168}{11},y=\frac{73}{11}$$