Whakaoti i te {l}{3x-y=11}{5x+3y=9}

$$\left\{ \begin{array} { l } { 3 x - y = 11 } \\ { 5 x + 3 y = 9 } \end{array} \right.$$

Whakaoti mō x, y

$$x=3$$

$$y=-2$$

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.

$$3x-y=11,5x+3y=9$$

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

$$3x-y=11$$

Me tāpiri y ki ngā taha e rua o te whārite.

$$3x=y+11$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(y+11\right)$$

Whakareatia \frac{1}{3} ki te y+11.

$$x=\frac{1}{3}y+\frac{11}{3}$$

Whakakapia te \frac{11+y}{3} mō te x ki tērā atu whārite, 5x+3y=9.

$$5\left(\frac{1}{3}y+\frac{11}{3}\right)+3y=9$$

Whakareatia 5 ki te \frac{11+y}{3}.

$$\frac{5}{3}y+\frac{55}{3}+3y=9$$

Tāpiri \frac{5y}{3} ki te 3y.

$$\frac{14}{3}y+\frac{55}{3}=9$$

Me tango \frac{55}{3} mai i 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{11}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{1}{3}\left(-2\right)+\frac{11}{3}$$

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

$$x=\frac{-2+11}{3}$$

Tāpiri \frac{11}{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.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=-2$$

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.

$$3x-y=11,5x+3y=9$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&-1\\5&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}11\\9\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}3&-1\\5&3\end{matrix}\right))\left(\begin{matrix}3&-1\\5&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-1\\5&3\end{matrix}\right))\left(\begin{matrix}11\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-1\\5&3\end{matrix}\right))\left(\begin{matrix}11\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-1\\5&3\end{matrix}\right))\left(\begin{matrix}11\\9\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{3\times 3-\left(-5\right)}&-\frac{-1}{3\times 3-\left(-5\right)}\\-\frac{5}{3\times 3-\left(-5\right)}&\frac{3}{3\times 3-\left(-5\right)}\end{matrix}\right)\left(\begin{matrix}11\\9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{14}&\frac{1}{14}\\-\frac{5}{14}&\frac{3}{14}\end{matrix}\right)\left(\begin{matrix}11\\9\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{14}\times 11+\frac{1}{14}\times 9\\-\frac{5}{14}\times 11+\frac{3}{14}\times 9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3\\-2\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=3,y=-2$$

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.

$$3x-y=11,5x+3y=9$$

Kia ōrite ai a 3x me 5x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 5 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$5\times 3x+5\left(-1\right)y=5\times 11,3\times 5x+3\times 3y=3\times 9$$

Whakarūnātia.

$$15x-5y=55,15x+9y=27$$

Me tango 15x+9y=27 mai i 15x-5y=55 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$15x-15x-5y-9y=55-27$$

Tāpiri 15x ki te -15x. Ka whakakore atu ngā kupu 15x me -15x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-5y-9y=55-27$$

Tāpiri -5y ki te -9y.

$$-14y=55-27$$

Tāpiri 55 ki te -27.

$$-14y=28$$

Whakawehea ngā taha e rua ki te -14.

$$y=-2$$

Whakaurua te -2 mō y ki 5x+3y=9. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$5x+3\left(-2\right)=9$$

Whakareatia 3 ki te -2.

$$5x-6=9$$

Me tāpiri 6 ki ngā taha e rua o te whārite.

$$5x=15$$

Whakawehea ngā taha e rua ki te 5.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=-2$$