Whakaoti i te {l}{3x+7y=10}{4x-19y=7}

$$\left. \begin{array} { l } { 3 x + 7 y = 10 } \\ { 4 x - 19 y = 7 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{239}{85} = 2\frac{69}{85} \approx 2.811764706$$

$$y=\frac{19}{85}\approx 0.223529412$$

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.

$$3x+7y=10,4x-19y=7$$

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

$$3x+7y=10$$

Me tango 7y mai i ngā taha e rua o te whārite.

$$3x=-7y+10$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(-7y+10\right)$$

Whakareatia \frac{1}{3} ki te -7y+10.

$$x=-\frac{7}{3}y+\frac{10}{3}$$

Whakakapia te \frac{-7y+10}{3} mō te x ki tērā atu whārite, 4x-19y=7.

$$4\left(-\frac{7}{3}y+\frac{10}{3}\right)-19y=7$$

Whakareatia 4 ki te \frac{-7y+10}{3}.

$$-\frac{28}{3}y+\frac{40}{3}-19y=7$$

Tāpiri -\frac{28y}{3} ki te -19y.

$$-\frac{85}{3}y+\frac{40}{3}=7$$

Me tango \frac{40}{3} mai i ngā taha e rua o te whārite.

$$-\frac{85}{3}y=-\frac{19}{3}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{85}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{19}{85}$$

Whakaurua te \frac{19}{85} mō y ki x=-\frac{7}{3}y+\frac{10}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{7}{3}\times \frac{19}{85}+\frac{10}{3}$$

Whakareatia -\frac{7}{3} ki te \frac{19}{85} 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.

$$x=-\frac{133}{255}+\frac{10}{3}$$

Tāpiri \frac{10}{3} ki te -\frac{133}{255} 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.

$$x=\frac{239}{85}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{239}{85},y=\frac{19}{85}$$

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.

$$3x+7y=10,4x-19y=7$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&7\\4&-19\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}10\\7\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&7\\4&-19\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&7\\4&-19\end{matrix}\right))\left(\begin{matrix}3&7\\4&-19\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&7\\4&-19\end{matrix}\right))\left(\begin{matrix}10\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&7\\4&-19\end{matrix}\right))\left(\begin{matrix}10\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&7\\4&-19\end{matrix}\right))\left(\begin{matrix}10\\7\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{19}{3\left(-19\right)-7\times 4}&-\frac{7}{3\left(-19\right)-7\times 4}\\-\frac{4}{3\left(-19\right)-7\times 4}&\frac{3}{3\left(-19\right)-7\times 4}\end{matrix}\right)\left(\begin{matrix}10\\7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{19}{85}&\frac{7}{85}\\\frac{4}{85}&-\frac{3}{85}\end{matrix}\right)\left(\begin{matrix}10\\7\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{19}{85}\times 10+\frac{7}{85}\times 7\\\frac{4}{85}\times 10-\frac{3}{85}\times 7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{239}{85}\\\frac{19}{85}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{239}{85},y=\frac{19}{85}$$

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.

$$3x+7y=10,4x-19y=7$$

Kia ōrite ai a 3x 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 3.

$$4\times 3x+4\times 7y=4\times 10,3\times 4x+3\left(-19\right)y=3\times 7$$

Whakarūnātia.

$$12x+28y=40,12x-57y=21$$

Me tango 12x-57y=21 mai i 12x+28y=40 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$12x-12x+28y+57y=40-21$$

Tāpiri 12x ki te -12x. Ka whakakore atu ngā kupu 12x me -12x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$28y+57y=40-21$$

Tāpiri 28y ki te 57y.

$$85y=40-21$$

Tāpiri 40 ki te -21.

$$85y=19$$

Whakawehea ngā taha e rua ki te 85.

$$y=\frac{19}{85}$$

Whakaurua te \frac{19}{85} mō y ki 4x-19y=7. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x-19\times \frac{19}{85}=7$$

Whakareatia -19 ki te \frac{19}{85}.

$$4x-\frac{361}{85}=7$$

Me tāpiri \frac{361}{85} ki ngā taha e rua o te whārite.

$$4x=\frac{956}{85}$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{239}{85}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{239}{85},y=\frac{19}{85}$$