Whakaoti i te {l}{5x-3y=18}{2x+y=5}

$$\left. \begin{array} { l } { 5 x - 3 y = 18 } \\ { 2 x + y = 5 } \end{array} \right.$$

Whakaoti mō x, y

$$x=3$$

$$y=-1$$

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.

$$5x-3y=18,2x+y=5$$

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

$$5x-3y=18$$

Me tāpiri 3y ki ngā taha e rua o te whārite.

$$5x=3y+18$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\left(3y+18\right)$$

Whakareatia \frac{1}{5} ki te 18+3y.

$$x=\frac{3}{5}y+\frac{18}{5}$$

Whakakapia te \frac{18+3y}{5} mō te x ki tērā atu whārite, 2x+y=5.

$$2\left(\frac{3}{5}y+\frac{18}{5}\right)+y=5$$

Whakareatia 2 ki te \frac{18+3y}{5}.

$$\frac{6}{5}y+\frac{36}{5}+y=5$$

Tāpiri \frac{6y}{5} ki te y.

$$\frac{11}{5}y+\frac{36}{5}=5$$

Me tango \frac{36}{5} mai i ngā taha e rua o te whārite.

$$\frac{11}{5}y=-\frac{11}{5}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{11}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-1$$

Whakaurua te -1 mō y ki x=\frac{3}{5}y+\frac{18}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{3}{5}\left(-1\right)+\frac{18}{5}$$

Whakareatia \frac{3}{5} ki te -1.

$$x=\frac{-3+18}{5}$$

Tāpiri \frac{18}{5} ki te -\frac{3}{5} 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.

$$x=3,y=-1$$

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.

$$5x-3y=18,2x+y=5$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&-3\\2&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}18\\5\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}5&-3\\2&1\end{matrix}\right))\left(\begin{matrix}5&-3\\2&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-3\\2&1\end{matrix}\right))\left(\begin{matrix}18\\5\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-3\\2&1\end{matrix}\right))\left(\begin{matrix}18\\5\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-3\\2&1\end{matrix}\right))\left(\begin{matrix}18\\5\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{5-\left(-3\times 2\right)}&-\frac{-3}{5-\left(-3\times 2\right)}\\-\frac{2}{5-\left(-3\times 2\right)}&\frac{5}{5-\left(-3\times 2\right)}\end{matrix}\right)\left(\begin{matrix}18\\5\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{11}&\frac{3}{11}\\-\frac{2}{11}&\frac{5}{11}\end{matrix}\right)\left(\begin{matrix}18\\5\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{11}\times 18+\frac{3}{11}\times 5\\-\frac{2}{11}\times 18+\frac{5}{11}\times 5\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=3,y=-1$$

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.

$$5x-3y=18,2x+y=5$$

Kia ōrite ai a 5x me 2x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 5.

$$2\times 5x+2\left(-3\right)y=2\times 18,5\times 2x+5y=5\times 5$$

Whakarūnātia.

$$10x-6y=36,10x+5y=25$$

Me tango 10x+5y=25 mai i 10x-6y=36 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$10x-10x-6y-5y=36-25$$

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.

$$-6y-5y=36-25$$

Tāpiri -6y ki te -5y.

$$-11y=36-25$$

Tāpiri 36 ki te -25.

$$-11y=11$$

Whakawehea ngā taha e rua ki te -11.

$$y=-1$$

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

$$2x-1=5$$

Me tāpiri 1 ki ngā taha e rua o te whārite.

$$2x=6$$

Whakawehea ngā taha e rua ki te 2.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=-1$$