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

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

Whakaoti mō y, x

$$x = \frac{5}{2} = 2\frac{1}{2} = 2.5$$

$$y=0$$

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.

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

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

$$5y+2x=5$$

Me tango 2x mai i ngā taha e rua o te whārite.

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

Whakawehea ngā taha e rua ki te 5.

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

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

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

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

$$-\frac{2}{5}x+1+2x=5$$

Tāpiri -\frac{2x}{5} ki te 2x.

$$\frac{8}{5}x+1=5$$

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

$$\frac{8}{5}x=4$$

Whakawehea ngā taha e rua o te whārite ki te \frac{8}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

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

Whakaurua te \frac{5}{2} mō x ki y=-\frac{2}{5}x+1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

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

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

$$y=-1+1$$

Tāpiri 1 ki te -1.

$$y=0$$

Kua oti te pūnaha te whakatau.

$$y=0,x=\frac{5}{2}$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&2\\1&2\end{matrix}\right).

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}&-\frac{1}{4}\\-\frac{1}{8}&\frac{5}{8}\end{matrix}\right)\left(\begin{matrix}5\\5\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}\times 5-\frac{1}{4}\times 5\\-\frac{1}{8}\times 5+\frac{5}{8}\times 5\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}0\\\frac{5}{2}\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=0,x=\frac{5}{2}$$

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.

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

Me tango y+2x=5 mai i 5y+2x=5 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

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

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.

$$5y-y=5-5$$

Tāpiri 5y ki te -y.

$$4y=5-5$$

Tāpiri 5 ki te -5.

$$4y=0$$

Whakawehea ngā taha e rua ki te 4.

$$y=0$$

Whakaurua te 0 mō y ki y+2x=5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x=5$$

Kua oti te pūnaha te whakatau.

$$y=0,x=\frac{5}{2}$$