Whakaoti i te {l}{3x+y=-1}{x+5y=9}

$$\left\{ \begin{array} { l } { 3 x + y = - 1 } \\ { x + 5 y = 9 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-1$$

$$y=2$$

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.

$$3x+y=-1,x+5y=9$$

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

$$3x+y=-1$$

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

$$3x=-y-1$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(-y-1\right)$$

Whakareatia \frac{1}{3} ki te -y-1.

$$x=-\frac{1}{3}y-\frac{1}{3}$$

Whakakapia te \frac{-y-1}{3} mō te x ki tērā atu whārite, x+5y=9.

$$-\frac{1}{3}y-\frac{1}{3}+5y=9$$

Tāpiri -\frac{y}{3} ki te 5y.

$$\frac{14}{3}y-\frac{1}{3}=9$$

Me tāpiri \frac{1}{3} ki ngā taha e rua o te whārite.

$$\frac{14}{3}y=\frac{28}{3}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{14}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=2$$

Whakaurua te 2 mō y ki x=-\frac{1}{3}y-\frac{1}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{3}\times 2-\frac{1}{3}$$

Whakareatia -\frac{1}{3} ki te 2.

$$x=\frac{-2-1}{3}$$

Tāpiri -\frac{1}{3} ki te -\frac{2}{3} 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.

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

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.

$$3x+y=-1,x+5y=9$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&1\\1&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-1\\9\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&1\\1&5\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&1\\1&5\end{matrix}\right))\left(\begin{matrix}3&1\\1&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&1\\1&5\end{matrix}\right))\left(\begin{matrix}-1\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&1\\1&5\end{matrix}\right))\left(\begin{matrix}-1\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&1\\1&5\end{matrix}\right))\left(\begin{matrix}-1\\9\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{3\times 5-1}&-\frac{1}{3\times 5-1}\\-\frac{1}{3\times 5-1}&\frac{3}{3\times 5-1}\end{matrix}\right)\left(\begin{matrix}-1\\9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{14}&-\frac{1}{14}\\-\frac{1}{14}&\frac{3}{14}\end{matrix}\right)\left(\begin{matrix}-1\\9\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{14}\left(-1\right)-\frac{1}{14}\times 9\\-\frac{1}{14}\left(-1\right)+\frac{3}{14}\times 9\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

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

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.

$$3x+y=-1,x+5y=9$$

Kia ōrite ai a 3x 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 3.

$$3x+y=-1,3x+3\times 5y=3\times 9$$

Whakarūnātia.

$$3x+y=-1,3x+15y=27$$

Me tango 3x+15y=27 mai i 3x+y=-1 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3x-3x+y-15y=-1-27$$

Tāpiri 3x ki te -3x. Ka whakakore atu ngā kupu 3x me -3x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$y-15y=-1-27$$

Tāpiri y ki te -15y.

$$-14y=-1-27$$

Tāpiri -1 ki te -27.

$$-14y=-28$$

Whakawehea ngā taha e rua ki te -14.

$$y=2$$

Whakaurua te 2 mō y ki x+5y=9. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x+5\times 2=9$$

Whakareatia 5 ki te 2.

$$x+10=9$$

Me tango 10 mai i ngā taha e rua o te whārite.

$$x=-1$$

Kua oti te pūnaha te whakatau.

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