Whakaoti i te {l}{6x+5y=1}{x-y=2}

$$\left\{ \begin{array} { l } { 6 x + 5 y = 1 } \\ { x - y = 2 } \end{array} \right.$$

Whakaoti mō x, y

$$x=1$$

$$y=-1$$

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.

$$6x+5y=1,x-y=2$$

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

$$6x+5y=1$$

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

$$6x=-5y+1$$

Whakawehea ngā taha e rua ki te 6.

$$x=\frac{1}{6}\left(-5y+1\right)$$

Whakareatia \frac{1}{6} ki te -5y+1.

$$x=-\frac{5}{6}y+\frac{1}{6}$$

Whakakapia te \frac{-5y+1}{6} mō te x ki tērā atu whārite, x-y=2.

$$-\frac{5}{6}y+\frac{1}{6}-y=2$$

Tāpiri -\frac{5y}{6} ki te -y.

$$-\frac{11}{6}y+\frac{1}{6}=2$$

Me tango \frac{1}{6} mai i ngā taha e rua o te whārite.

$$-\frac{11}{6}y=\frac{11}{6}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{11}{6}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-1$$

Whakaurua te -1 mō y ki x=-\frac{5}{6}y+\frac{1}{6}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{5}{6}\left(-1\right)+\frac{1}{6}$$

Whakareatia -\frac{5}{6} ki te -1.

$$x=\frac{5+1}{6}$$

Tāpiri \frac{1}{6} ki te \frac{5}{6} 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.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=-1$$

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.

$$6x+5y=1,x-y=2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}6&5\\1&-1\end{matrix}\right).

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

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

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

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

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

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{6\left(-1\right)-5}&-\frac{5}{6\left(-1\right)-5}\\-\frac{1}{6\left(-1\right)-5}&\frac{6}{6\left(-1\right)-5}\end{matrix}\right)\left(\begin{matrix}1\\2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{11}&\frac{5}{11}\\\frac{1}{11}&-\frac{6}{11}\end{matrix}\right)\left(\begin{matrix}1\\2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{11}+\frac{5}{11}\times 2\\\frac{1}{11}-\frac{6}{11}\times 2\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=1,y=-1$$

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.

$$6x+5y=1,x-y=2$$

Kia ōrite ai a 6x 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 6.

$$6x+5y=1,6x+6\left(-1\right)y=6\times 2$$

Whakarūnātia.

$$6x+5y=1,6x-6y=12$$

Me tango 6x-6y=12 mai i 6x+5y=1 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x+5y+6y=1-12$$

Tāpiri 6x ki te -6x. Ka whakakore atu ngā kupu 6x me -6x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$5y+6y=1-12$$

Tāpiri 5y ki te 6y.

$$11y=1-12$$

Tāpiri 1 ki te -12.

$$11y=-11$$

Whakawehea ngā taha e rua ki te 11.

$$y=-1$$

Whakaurua te -1 mō y ki x-y=2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x-\left(-1\right)=2$$

Me tango 1 mai i ngā taha e rua o te whārite.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=-1$$