Whakaoti i te {l}{2x-3y=1}{3x+5y=1}

$$\left\{ \begin{array} { l } { 2 x - 3 y = 1 } \\ { 3 x + 5 y = 1 } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{8}{19}\approx 0.421052632$$

$$y=-\frac{1}{19}\approx -0.052631579$$

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.

$$2x-3y=1,3x+5y=1$$

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

$$2x-3y=1$$

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

$$2x=3y+1$$

Whakawehea ngā taha e rua ki te 2.

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

Whakareatia \frac{1}{2} ki te 3y+1.

$$x=\frac{3}{2}y+\frac{1}{2}$$

Whakakapia te \frac{3y+1}{2} mō te x ki tērā atu whārite, 3x+5y=1.

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

Whakareatia 3 ki te \frac{3y+1}{2}.

$$\frac{9}{2}y+\frac{3}{2}+5y=1$$

Tāpiri \frac{9y}{2} ki te 5y.

$$\frac{19}{2}y+\frac{3}{2}=1$$

Me tango \frac{3}{2} mai i ngā taha e rua o te whārite.

$$\frac{19}{2}y=-\frac{1}{2}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{19}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-\frac{1}{19}$$

Whakaurua te -\frac{1}{19} mō y ki x=\frac{3}{2}y+\frac{1}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{3}{2}\left(-\frac{1}{19}\right)+\frac{1}{2}$$

Whakareatia \frac{3}{2} ki te -\frac{1}{19} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=-\frac{3}{38}+\frac{1}{2}$$

Tāpiri \frac{1}{2} ki te -\frac{3}{38} 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.

$$x=\frac{8}{19}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{8}{19},y=-\frac{1}{19}$$

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.

$$2x-3y=1,3x+5y=1$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{19}&\frac{3}{19}\\-\frac{3}{19}&\frac{2}{19}\end{matrix}\right)\left(\begin{matrix}1\\1\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5+3}{19}\\\frac{-3+2}{19}\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{8}{19}\\-\frac{1}{19}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{8}{19},y=-\frac{1}{19}$$

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.

$$2x-3y=1,3x+5y=1$$

Kia ōrite ai a 2x me 3x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 3 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

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

Whakarūnātia.

$$6x-9y=3,6x+10y=2$$

Me tango 6x+10y=2 mai i 6x-9y=3 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x-9y-10y=3-2$$

Tāpiri 6x ki te -6x. Ka whakakore atu ngā kupu 6x me -6x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-9y-10y=3-2$$

Tāpiri -9y ki te -10y.

$$-19y=3-2$$

Tāpiri 3 ki te -2.

$$-19y=1$$

Whakawehea ngā taha e rua ki te -19.

$$y=-\frac{1}{19}$$

Whakaurua te -\frac{1}{19} mō y ki 3x+5y=1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x+5\left(-\frac{1}{19}\right)=1$$

Whakareatia 5 ki te -\frac{1}{19}.

$$3x-\frac{5}{19}=1$$

Me tāpiri \frac{5}{19} ki ngā taha e rua o te whārite.

$$3x=\frac{24}{19}$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{8}{19}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{8}{19},y=-\frac{1}{19}$$