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

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

Whakaoti mō x, y

$$x=\frac{1}{5}=0.2$$

$$y=0$$

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.

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

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

$$5x+6y=1$$

Me tango 6y mai i ngā taha e rua o te whārite.

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

Whakawehea ngā taha e rua ki te 5.

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

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

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

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

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

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

$$-6y+1+y=1$$

Tāpiri -6y ki te y.

$$-5y+1=1$$

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

$$-5y=0$$

Whakawehea ngā taha e rua ki te -5.

$$y=0$$

Whakaurua te 0 mō y ki x=-\frac{6}{5}y+\frac{1}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

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

Kua oti te pūnaha te whakatau.

$$x=\frac{1}{5},y=0$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{25}&\frac{6}{25}\\\frac{1}{5}&-\frac{1}{5}\end{matrix}\right)\left(\begin{matrix}1\\1\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{-1+6}{25}\\\frac{1-1}{5}\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{5}\\0\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{1}{5},y=0$$

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.

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

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

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

Tāpiri 5x ki te -5x. Ka whakakore atu ngā kupu 5x me -5x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$6y-y=1-1$$

Tāpiri 6y ki te -y.

$$5y=1-1$$

Tāpiri 1 ki te -1.

$$5y=0$$

Whakawehea ngā taha e rua ki te 5.

$$y=0$$

Whakaurua te 0 mō y ki 5x+y=1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$5x=1$$

Whakawehea ngā taha e rua ki te 5.

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

Kua oti te pūnaha te whakatau.

$$x=\frac{1}{5},y=0$$