Whakaoti i te {c}{x+y=-0.5}{-0.6x+0.7y=0.82}

$$\left. \begin{array} { c } { x + y = - 0.5 } \\ { - 0.6 x + 0.7 y = 0.82 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-0.9$$

$$y=0.4$$

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.

$$x+y=-0.5,-0.6x+0.7y=0.82$$

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

$$x+y=-0.5$$

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

$$x=-y-0.5$$

Whakakapia te -y-0.5 mō te x ki tērā atu whārite, -0.6x+0.7y=0.82.

$$-0.6\left(-y-0.5\right)+0.7y=0.82$$

Whakareatia -0.6 ki te -y-0.5.

$$0.6y+0.3+0.7y=0.82$$

Tāpiri \frac{3y}{5} ki te \frac{7y}{10}.

$$1.3y+0.3=0.82$$

Me tango 0.3 mai i ngā taha e rua o te whārite.

$$1.3y=0.52$$

Whakawehea ngā taha e rua o te whārite ki te 1.3, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=0.4$$

Whakaurua te 0.4 mō y ki x=-y-0.5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-0.4-0.5$$

Tāpiri -0.5 ki te -0.4 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.

$$x=-0.9$$

Kua oti te pūnaha te whakatau.

$$x=-0.9,y=0.4$$

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.

$$x+y=-0.5,-0.6x+0.7y=0.82$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\-0.6&0.7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-0.5\\0.82\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&1\\-0.6&0.7\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&1\\-0.6&0.7\end{matrix}\right))\left(\begin{matrix}1&1\\-0.6&0.7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\-0.6&0.7\end{matrix}\right))\left(\begin{matrix}-0.5\\0.82\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\-0.6&0.7\end{matrix}\right))\left(\begin{matrix}-0.5\\0.82\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\-0.6&0.7\end{matrix}\right))\left(\begin{matrix}-0.5\\0.82\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{0.7}{0.7-\left(-0.6\right)}&-\frac{1}{0.7-\left(-0.6\right)}\\-\frac{-0.6}{0.7-\left(-0.6\right)}&\frac{1}{0.7-\left(-0.6\right)}\end{matrix}\right)\left(\begin{matrix}-0.5\\0.82\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{13}&-\frac{10}{13}\\\frac{6}{13}&\frac{10}{13}\end{matrix}\right)\left(\begin{matrix}-0.5\\0.82\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{13}\left(-0.5\right)-\frac{10}{13}\times 0.82\\\frac{6}{13}\left(-0.5\right)+\frac{10}{13}\times 0.82\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-0.9\\0.4\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-0.9,y=0.4$$

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.

$$x+y=-0.5,-0.6x+0.7y=0.82$$

Kia ōrite ai a x me -\frac{3x}{5}, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -0.6 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$-0.6x-0.6y=-0.6\left(-0.5\right),-0.6x+0.7y=0.82$$

Whakarūnātia.

$$-0.6x-0.6y=0.3,-0.6x+0.7y=0.82$$

Me tango -0.6x+0.7y=0.82 mai i -0.6x-0.6y=0.3 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-0.6x+0.6x-0.6y-0.7y=0.3-0.82$$

Tāpiri -\frac{3x}{5} ki te \frac{3x}{5}. Ka whakakore atu ngā kupu -\frac{3x}{5} me \frac{3x}{5}, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-0.6y-0.7y=0.3-0.82$$

Tāpiri -\frac{3y}{5} ki te -\frac{7y}{10}.

$$-1.3y=0.3-0.82$$

Tāpiri 0.3 ki te -0.82 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.

$$-1.3y=-0.52$$

Whakawehea ngā taha e rua o te whārite ki te -1.3, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=0.4$$

Whakaurua te 0.4 mō y ki -0.6x+0.7y=0.82. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-0.6x+0.7\times 0.4=0.82$$

Whakareatia 0.7 ki te 0.4 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.

$$-0.6x+0.28=0.82$$

Me tango 0.28 mai i ngā taha e rua o te whārite.

$$-0.6x=0.54$$

Whakawehea ngā taha e rua o te whārite ki te -0.6, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=-0.9$$

Kua oti te pūnaha te whakatau.

$$x=-0.9,y=0.4$$