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

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

Whakaoti mō x, y

$$x = \frac{27}{7} = 3\frac{6}{7} \approx 3.857142857$$

$$y=\frac{4}{7}\approx 0.571428571$$

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.

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

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

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

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

$$-x=-5y-1$$

Whakawehea ngā taha e rua ki te -1.

$$x=-\left(-5y-1\right)$$

Whakareatia -1 ki te -5y-1.

$$x=5y+1$$

Whakakapia te 5y+1 mō te x ki tērā atu whārite, x+2y=5.

$$5y+1+2y=5$$

Tāpiri 5y ki te 2y.

$$7y+1=5$$

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

$$7y=4$$

Whakawehea ngā taha e rua ki te 7.

$$y=\frac{4}{7}$$

Whakaurua te \frac{4}{7} mō y ki x=5y+1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=5\times \frac{4}{7}+1$$

Whakareatia 5 ki te \frac{4}{7}.

$$x=\frac{20}{7}+1$$

Tāpiri 1 ki te \frac{20}{7}.

$$x=\frac{27}{7}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{27}{7},y=\frac{4}{7}$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{7}&\frac{5}{7}\\\frac{1}{7}&\frac{1}{7}\end{matrix}\right)\left(\begin{matrix}-1\\5\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{7}\left(-1\right)+\frac{5}{7}\times 5\\\frac{1}{7}\left(-1\right)+\frac{1}{7}\times 5\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{27}{7}\\\frac{4}{7}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{27}{7},y=\frac{4}{7}$$

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.

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

Kia ōrite ai a -x 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 -1.

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

Me tango -x-2y=-5 mai i -x+5y=-1 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

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

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.

$$5y+2y=-1+5$$

Tāpiri 5y ki te 2y.

$$7y=-1+5$$

Tāpiri -1 ki te 5.

$$7y=4$$

Whakawehea ngā taha e rua ki te 7.

$$y=\frac{4}{7}$$

Whakaurua te \frac{4}{7} mō y ki x+2y=5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x+2\times \frac{4}{7}=5$$

Whakareatia 2 ki te \frac{4}{7}.

$$x+\frac{8}{7}=5$$

Me tango \frac{8}{7} mai i ngā taha e rua o te whārite.

$$x=\frac{27}{7}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{27}{7},y=\frac{4}{7}$$