Whakaoti i te {c}{-2x+7y=10}{3x+7y=2}

$$\left. \begin{array} { c } { - 2 x + 7 y = 10 } \\ { 3 x + 7 y = 2 } \end{array} \right.$$

Whakaoti mō x, y

$$x = -\frac{8}{5} = -1\frac{3}{5} = -1.6$$

$$y=\frac{34}{35}\approx 0.971428571$$

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.

$$-2x+7y=10,3x+7y=2$$

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

$$-2x+7y=10$$

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

$$-2x=-7y+10$$

Whakawehea ngā taha e rua ki te -2.

$$x=-\frac{1}{2}\left(-7y+10\right)$$

Whakareatia -\frac{1}{2} ki te -7y+10.

$$x=\frac{7}{2}y-5$$

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

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

Whakareatia 3 ki te \frac{7y}{2}-5.

$$\frac{21}{2}y-15+7y=2$$

Tāpiri \frac{21y}{2} ki te 7y.

$$\frac{35}{2}y-15=2$$

Me tāpiri 15 ki ngā taha e rua o te whārite.

$$\frac{35}{2}y=17$$

Whakawehea ngā taha e rua o te whārite ki te \frac{35}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{34}{35}$$

Whakaurua te \frac{34}{35} mō y ki x=\frac{7}{2}y-5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{7}{2}\times \frac{34}{35}-5$$

Whakareatia \frac{7}{2} ki te \frac{34}{35} 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.

$$x=\frac{17}{5}-5$$

Tāpiri -5 ki te \frac{17}{5}.

$$x=-\frac{8}{5}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{8}{5},y=\frac{34}{35}$$

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.

$$-2x+7y=10,3x+7y=2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-2&7\\3&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}10\\2\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-2&7\\3&7\end{matrix}\right).

$$inverse(\left(\begin{matrix}-2&7\\3&7\end{matrix}\right))\left(\begin{matrix}-2&7\\3&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-2&7\\3&7\end{matrix}\right))\left(\begin{matrix}10\\2\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-2&7\\3&7\end{matrix}\right))\left(\begin{matrix}10\\2\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-2&7\\3&7\end{matrix}\right))\left(\begin{matrix}10\\2\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{-2\times 7-7\times 3}&-\frac{7}{-2\times 7-7\times 3}\\-\frac{3}{-2\times 7-7\times 3}&-\frac{2}{-2\times 7-7\times 3}\end{matrix}\right)\left(\begin{matrix}10\\2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{5}&\frac{1}{5}\\\frac{3}{35}&\frac{2}{35}\end{matrix}\right)\left(\begin{matrix}10\\2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{5}\times 10+\frac{1}{5}\times 2\\\frac{3}{35}\times 10+\frac{2}{35}\times 2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{8}{5}\\\frac{34}{35}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{8}{5},y=\frac{34}{35}$$

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.

$$-2x+7y=10,3x+7y=2$$

Me tango 3x+7y=2 mai i -2x+7y=10 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-2x-3x+7y-7y=10-2$$

Tāpiri 7y ki te -7y. Ka whakakore atu ngā kupu 7y me -7y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-2x-3x=10-2$$

Tāpiri -2x ki te -3x.

$$-5x=10-2$$

Tāpiri 10 ki te -2.

$$-5x=8$$

Whakawehea ngā taha e rua ki te -5.

$$x=-\frac{8}{5}$$

Whakaurua te -\frac{8}{5} mō x ki 3x+7y=2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$3\left(-\frac{8}{5}\right)+7y=2$$

Whakareatia 3 ki te -\frac{8}{5}.

$$-\frac{24}{5}+7y=2$$

Me tāpiri \frac{24}{5} ki ngā taha e rua o te whārite.

$$7y=\frac{34}{5}$$

Whakawehea ngā taha e rua ki te 7.

$$y=\frac{34}{35}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{8}{5},y=\frac{34}{35}$$