Whakaoti i te {l}{x-7y=6}{5x+3y=2}

$$\left\{ \begin{array} { l } { x - 7 y = 6 } \\ { 5 x + 3 y = 2 } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{16}{19}\approx 0.842105263$$

$$y=-\frac{14}{19}\approx -0.736842105$$

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.

$$x-7y=6,5x+3y=2$$

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

$$x-7y=6$$

Me tāpiri 7y ki ngā taha e rua o te whārite.

$$x=7y+6$$

Whakakapia te 7y+6 mō te x ki tērā atu whārite, 5x+3y=2.

$$5\left(7y+6\right)+3y=2$$

Whakareatia 5 ki te 7y+6.

$$35y+30+3y=2$$

Tāpiri 35y ki te 3y.

$$38y+30=2$$

Me tango 30 mai i ngā taha e rua o te whārite.

$$38y=-28$$

Whakawehea ngā taha e rua ki te 38.

$$y=-\frac{14}{19}$$

Whakaurua te -\frac{14}{19} mō y ki x=7y+6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=7\left(-\frac{14}{19}\right)+6$$

Whakareatia 7 ki te -\frac{14}{19}.

$$x=-\frac{98}{19}+6$$

Tāpiri 6 ki te -\frac{98}{19}.

$$x=\frac{16}{19}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{16}{19},y=-\frac{14}{19}$$

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.

$$x-7y=6,5x+3y=2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-7\\5&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}6\\2\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-7\\5&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-7\\5&3\end{matrix}\right))\left(\begin{matrix}1&-7\\5&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-7\\5&3\end{matrix}\right))\left(\begin{matrix}6\\2\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-7\\5&3\end{matrix}\right))\left(\begin{matrix}6\\2\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-7\\5&3\end{matrix}\right))\left(\begin{matrix}6\\2\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{3-\left(-7\times 5\right)}&-\frac{-7}{3-\left(-7\times 5\right)}\\-\frac{5}{3-\left(-7\times 5\right)}&\frac{1}{3-\left(-7\times 5\right)}\end{matrix}\right)\left(\begin{matrix}6\\2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{38}&\frac{7}{38}\\-\frac{5}{38}&\frac{1}{38}\end{matrix}\right)\left(\begin{matrix}6\\2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{38}\times 6+\frac{7}{38}\times 2\\-\frac{5}{38}\times 6+\frac{1}{38}\times 2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{16}{19}\\-\frac{14}{19}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{16}{19},y=-\frac{14}{19}$$

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.

$$x-7y=6,5x+3y=2$$

Kia ōrite ai a x me 5x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 5 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$5x+5\left(-7\right)y=5\times 6,5x+3y=2$$

Whakarūnātia.

$$5x-35y=30,5x+3y=2$$

Me tango 5x+3y=2 mai i 5x-35y=30 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$5x-5x-35y-3y=30-2$$

Tāpiri 5x ki te -5x. Ka whakakore atu ngā kupu 5x me -5x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-35y-3y=30-2$$

Tāpiri -35y ki te -3y.

$$-38y=30-2$$

Tāpiri 30 ki te -2.

$$-38y=28$$

Whakawehea ngā taha e rua ki te -38.

$$y=-\frac{14}{19}$$

Whakaurua te -\frac{14}{19} mō y ki 5x+3y=2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$5x+3\left(-\frac{14}{19}\right)=2$$

Whakareatia 3 ki te -\frac{14}{19}.

$$5x-\frac{42}{19}=2$$

Me tāpiri \frac{42}{19} ki ngā taha e rua o te whārite.

$$5x=\frac{80}{19}$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{16}{19}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{16}{19},y=-\frac{14}{19}$$