Whakaoti i te {l}{x+4y=6}{x-y=4}

$$\left. \begin{array} { l } { x + 4 y = 6 } \\ { x - y = 4 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{22}{5} = 4\frac{2}{5} = 4.4$$

$$y=\frac{2}{5}=0.4$$

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.

$$x+4y=6,x-y=4$$

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

$$x+4y=6$$

Me tango 4y mai i ngā taha e rua o te whārite.

$$x=-4y+6$$

Whakakapia te -4y+6 mō te x ki tērā atu whārite, x-y=4.

$$-4y+6-y=4$$

Tāpiri -4y ki te -y.

$$-5y+6=4$$

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

$$-5y=-2$$

Whakawehea ngā taha e rua ki te -5.

$$y=\frac{2}{5}$$

Whakaurua te \frac{2}{5} mō y ki x=-4y+6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-4\times \frac{2}{5}+6$$

Whakareatia -4 ki te \frac{2}{5}.

$$x=-\frac{8}{5}+6$$

Tāpiri 6 ki te -\frac{8}{5}.

$$x=\frac{22}{5}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{22}{5},y=\frac{2}{5}$$

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.

$$x+4y=6,x-y=4$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&4\\1&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}6\\4\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&4\\1&-1\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&4\\1&-1\end{matrix}\right))\left(\begin{matrix}1&4\\1&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&4\\1&-1\end{matrix}\right))\left(\begin{matrix}6\\4\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&4\\1&-1\end{matrix}\right))\left(\begin{matrix}6\\4\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&4\\1&-1\end{matrix}\right))\left(\begin{matrix}6\\4\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{-1-4}&-\frac{4}{-1-4}\\-\frac{1}{-1-4}&\frac{1}{-1-4}\end{matrix}\right)\left(\begin{matrix}6\\4\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{5}&\frac{4}{5}\\\frac{1}{5}&-\frac{1}{5}\end{matrix}\right)\left(\begin{matrix}6\\4\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{5}\times 6+\frac{4}{5}\times 4\\\frac{1}{5}\times 6-\frac{1}{5}\times 4\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{22}{5}\\\frac{2}{5}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{22}{5},y=\frac{2}{5}$$

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.

$$x+4y=6,x-y=4$$

Me tango x-y=4 mai i x+4y=6 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$x-x+4y+y=6-4$$

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

$$4y+y=6-4$$

Tāpiri 4y ki te y.

$$5y=6-4$$

Tāpiri 6 ki te -4.

$$5y=2$$

Whakawehea ngā taha e rua ki te 5.

$$y=\frac{2}{5}$$

Whakaurua te \frac{2}{5} mō y ki x-y=4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x-\frac{2}{5}=4$$

Me tāpiri \frac{2}{5} ki ngā taha e rua o te whārite.

$$x=\frac{22}{5}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{22}{5},y=\frac{2}{5}$$