Whakaoti i te {l}{2x+y=11}{5x+3y=30}

$$\left\{ \begin{array} { l } { 2 x + y = 11 } \\ { 5 x + 3 y = 30 } \end{array} \right.$$

Whakaoti mō x, y

$$x=3$$

$$y=5$$

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.

$$2x+y=11,5x+3y=30$$

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

$$2x+y=11$$

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

$$2x=-y+11$$

Whakawehea ngā taha e rua ki te 2.

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

Whakareatia \frac{1}{2} ki te -y+11.

$$x=-\frac{1}{2}y+\frac{11}{2}$$

Whakakapia te \frac{-y+11}{2} mō te x ki tērā atu whārite, 5x+3y=30.

$$5\left(-\frac{1}{2}y+\frac{11}{2}\right)+3y=30$$

Whakareatia 5 ki te \frac{-y+11}{2}.

$$-\frac{5}{2}y+\frac{55}{2}+3y=30$$

Tāpiri -\frac{5y}{2} ki te 3y.

$$\frac{1}{2}y+\frac{55}{2}=30$$

Me tango \frac{55}{2} mai i ngā taha e rua o te whārite.

$$\frac{1}{2}y=\frac{5}{2}$$

Me whakarea ngā taha e rua ki te 2.

$$y=5$$

Whakaurua te 5 mō y ki x=-\frac{1}{2}y+\frac{11}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{2}\times 5+\frac{11}{2}$$

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

$$x=\frac{-5+11}{2}$$

Tāpiri \frac{11}{2} ki te -\frac{5}{2} 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.

$$x=3,y=5$$

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.

$$2x+y=11,5x+3y=30$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&1\\5&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}11\\30\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&1\\5&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&1\\5&3\end{matrix}\right))\left(\begin{matrix}2&1\\5&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\5&3\end{matrix}\right))\left(\begin{matrix}11\\30\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\5&3\end{matrix}\right))\left(\begin{matrix}11\\30\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\5&3\end{matrix}\right))\left(\begin{matrix}11\\30\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{2\times 3-5}&-\frac{1}{2\times 3-5}\\-\frac{5}{2\times 3-5}&\frac{2}{2\times 3-5}\end{matrix}\right)\left(\begin{matrix}11\\30\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3&-1\\-5&2\end{matrix}\right)\left(\begin{matrix}11\\30\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3\times 11-30\\-5\times 11+2\times 30\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=3,y=5$$

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.

$$2x+y=11,5x+3y=30$$

Kia ōrite ai a 2x 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 2.

$$5\times 2x+5y=5\times 11,2\times 5x+2\times 3y=2\times 30$$

Whakarūnātia.

$$10x+5y=55,10x+6y=60$$

Me tango 10x+6y=60 mai i 10x+5y=55 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$10x-10x+5y-6y=55-60$$

Tāpiri 10x ki te -10x. Ka whakakore atu ngā kupu 10x me -10x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$5y-6y=55-60$$

Tāpiri 5y ki te -6y.

$$-y=55-60$$

Tāpiri 55 ki te -60.

$$-y=-5$$

Whakawehea ngā taha e rua ki te -1.

$$y=5$$

Whakaurua te 5 mō y ki 5x+3y=30. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$5x+3\times 5=30$$

Whakareatia 3 ki te 5.

$$5x+15=30$$

Me tango 15 mai i 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.

$$x=3,y=5$$