Whakaoti i te {l}{5x-3y-2=0}{4x+7y+3=0}

$$\left. \begin{array} { l } { 5 x - 3 y - 2 = 0 } \\ { 4 x + 7 y + 3 = 0 } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{5}{47}\approx 0.106382979$$

$$y=-\frac{23}{47}\approx -0.489361702$$

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.

$$5x-3y-2=0,4x+7y+3=0$$

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

$$5x-3y-2=0$$

Me tāpiri 2 ki ngā taha e rua o te whārite.

$$5x-3y=2$$

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

$$5x=3y+2$$

Whakawehea ngā taha e rua ki te 5.

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

Whakareatia \frac{1}{5} ki te 3y+2.

$$x=\frac{3}{5}y+\frac{2}{5}$$

Whakakapia te \frac{3y+2}{5} mō te x ki tērā atu whārite, 4x+7y+3=0.

$$4\left(\frac{3}{5}y+\frac{2}{5}\right)+7y+3=0$$

Whakareatia 4 ki te \frac{3y+2}{5}.

$$\frac{12}{5}y+\frac{8}{5}+7y+3=0$$

Tāpiri \frac{12y}{5} ki te 7y.

$$\frac{47}{5}y+\frac{8}{5}+3=0$$

Tāpiri \frac{8}{5} ki te 3.

$$\frac{47}{5}y+\frac{23}{5}=0$$

Me tango \frac{23}{5} mai i ngā taha e rua o te whārite.

$$\frac{47}{5}y=-\frac{23}{5}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{47}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-\frac{23}{47}$$

Whakaurua te -\frac{23}{47} mō y ki x=\frac{3}{5}y+\frac{2}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{3}{5}\left(-\frac{23}{47}\right)+\frac{2}{5}$$

Whakareatia \frac{3}{5} ki te -\frac{23}{47} 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.

$$x=-\frac{69}{235}+\frac{2}{5}$$

Tāpiri \frac{2}{5} ki te -\frac{69}{235} 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.

$$x=\frac{5}{47}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{5}{47},y=-\frac{23}{47}$$

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.

$$5x-3y-2=0,4x+7y+3=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&-3\\4&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2\\-3\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&-3\\4&7\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&-3\\4&7\end{matrix}\right))\left(\begin{matrix}5&-3\\4&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-3\\4&7\end{matrix}\right))\left(\begin{matrix}2\\-3\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-3\\4&7\end{matrix}\right))\left(\begin{matrix}2\\-3\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-3\\4&7\end{matrix}\right))\left(\begin{matrix}2\\-3\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te poukapa kōaro ko \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), kia tuhia anō ai te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{5\times 7-\left(-3\times 4\right)}&-\frac{-3}{5\times 7-\left(-3\times 4\right)}\\-\frac{4}{5\times 7-\left(-3\times 4\right)}&\frac{5}{5\times 7-\left(-3\times 4\right)}\end{matrix}\right)\left(\begin{matrix}2\\-3\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{47}&\frac{3}{47}\\-\frac{4}{47}&\frac{5}{47}\end{matrix}\right)\left(\begin{matrix}2\\-3\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{47}\times 2+\frac{3}{47}\left(-3\right)\\-\frac{4}{47}\times 2+\frac{5}{47}\left(-3\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{47}\\-\frac{23}{47}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{5}{47},y=-\frac{23}{47}$$

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.

$$5x-3y-2=0,4x+7y+3=0$$

Kia ōrite ai a 5x 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 5.

$$4\times 5x+4\left(-3\right)y+4\left(-2\right)=0,5\times 4x+5\times 7y+5\times 3=0$$

Whakarūnātia.

$$20x-12y-8=0,20x+35y+15=0$$

Me tango 20x+35y+15=0 mai i 20x-12y-8=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$20x-20x-12y-35y-8-15=0$$

Tāpiri 20x ki te -20x. Ka whakakore atu ngā kupu 20x me -20x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-12y-35y-8-15=0$$

Tāpiri -12y ki te -35y.

$$-47y-8-15=0$$

Tāpiri -8 ki te -15.

$$-47y-23=0$$

Me tāpiri 23 ki ngā taha e rua o te whārite.

$$-47y=23$$

Whakawehea ngā taha e rua ki te -47.

$$y=-\frac{23}{47}$$

Whakaurua te -\frac{23}{47} mō y ki 4x+7y+3=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x+7\left(-\frac{23}{47}\right)+3=0$$

Whakareatia 7 ki te -\frac{23}{47}.

$$4x-\frac{161}{47}+3=0$$

Tāpiri -\frac{161}{47} ki te 3.

$$4x-\frac{20}{47}=0$$

Me tāpiri \frac{20}{47} ki ngā taha e rua o te whārite.

$$4x=\frac{20}{47}$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{5}{47}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{5}{47},y=-\frac{23}{47}$$