Whakaoti i te {l}{2x+5y=1}{4x-3y=6}

$$\left\{ \begin{array} { l } { 2 x + 5 y = 1 } \\ { 4 x - 3 y = 6 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{33}{26} = 1\frac{7}{26} \approx 1.269230769$$

$$y=-\frac{4}{13}\approx -0.307692308$$

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.

$$2x+5y=1,4x-3y=6$$

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

$$2x+5y=1$$

Me tango 5y mai i ngā taha e rua o te whārite.

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

Whakawehea ngā taha e rua ki te 2.

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

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

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

Whakakapia te \frac{-5y+1}{2} mō te x ki tērā atu whārite, 4x-3y=6.

$$4\left(-\frac{5}{2}y+\frac{1}{2}\right)-3y=6$$

Whakareatia 4 ki te \frac{-5y+1}{2}.

$$-10y+2-3y=6$$

Tāpiri -10y ki te -3y.

$$-13y+2=6$$

Me tango 2 mai i ngā taha e rua o te whārite.

$$-13y=4$$

Whakawehea ngā taha e rua ki te -13.

$$y=-\frac{4}{13}$$

Whakaurua te -\frac{4}{13} mō y ki x=-\frac{5}{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.

$$x=-\frac{5}{2}\left(-\frac{4}{13}\right)+\frac{1}{2}$$

Whakareatia -\frac{5}{2} ki te -\frac{4}{13} 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.

$$x=\frac{10}{13}+\frac{1}{2}$$

Tāpiri \frac{1}{2} ki te \frac{10}{13} 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.

$$x=\frac{33}{26}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{33}{26},y=-\frac{4}{13}$$

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.

$$2x+5y=1,4x-3y=6$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&5\\4&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1\\6\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&5\\4&-3\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&5\\4&-3\end{matrix}\right))\left(\begin{matrix}2&5\\4&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&5\\4&-3\end{matrix}\right))\left(\begin{matrix}1\\6\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&5\\4&-3\end{matrix}\right))\left(\begin{matrix}1\\6\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&5\\4&-3\end{matrix}\right))\left(\begin{matrix}1\\6\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{2\left(-3\right)-5\times 4}&-\frac{5}{2\left(-3\right)-5\times 4}\\-\frac{4}{2\left(-3\right)-5\times 4}&\frac{2}{2\left(-3\right)-5\times 4}\end{matrix}\right)\left(\begin{matrix}1\\6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{26}&\frac{5}{26}\\\frac{2}{13}&-\frac{1}{13}\end{matrix}\right)\left(\begin{matrix}1\\6\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{26}+\frac{5}{26}\times 6\\\frac{2}{13}-\frac{1}{13}\times 6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{33}{26}\\-\frac{4}{13}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{33}{26},y=-\frac{4}{13}$$

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.

$$2x+5y=1,4x-3y=6$$

Kia ōrite ai a 2x me 4x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 4 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

$$4\times 2x+4\times 5y=4,2\times 4x+2\left(-3\right)y=2\times 6$$

Whakarūnātia.

$$8x+20y=4,8x-6y=12$$

Me tango 8x-6y=12 mai i 8x+20y=4 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$8x-8x+20y+6y=4-12$$

Tāpiri 8x ki te -8x. Ka whakakore atu ngā kupu 8x me -8x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$20y+6y=4-12$$

Tāpiri 20y ki te 6y.

$$26y=4-12$$

Tāpiri 4 ki te -12.

$$26y=-8$$

Whakawehea ngā taha e rua ki te 26.

$$y=-\frac{4}{13}$$

Whakaurua te -\frac{4}{13} mō y ki 4x-3y=6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x-3\left(-\frac{4}{13}\right)=6$$

Whakareatia -3 ki te -\frac{4}{13}.

$$4x+\frac{12}{13}=6$$

Me tango \frac{12}{13} mai i ngā taha e rua o te whārite.

$$4x=\frac{66}{13}$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{33}{26}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{33}{26},y=-\frac{4}{13}$$