Whakaoti i te {l}{-0.1x-0.7y-610=0}{-0.8x+0.5y+920=0}

$$\left. \begin{array} { l } { - 0.1 x - 0.7 y - 610 = 0 } \\ { - 0.8 x + 0.5 y + 920 = 0 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{33900}{61} = 555\frac{45}{61} \approx 555.737704918$$

$$y = -\frac{58000}{61} = -950\frac{50}{61} \approx -950.819672131$$

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.

$$-0.1x-0.7y-610=0,-0.8x+0.5y+920=0$$

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

$$-0.1x-0.7y-610=0$$

Me tāpiri 610 ki ngā taha e rua o te whārite.

$$-0.1x-0.7y=610$$

Me tāpiri \frac{7y}{10} ki ngā taha e rua o te whārite.

$$-0.1x=0.7y+610$$

Me whakarea ngā taha e rua ki te -10.

$$x=-10\left(0.7y+610\right)$$

Whakareatia -10 ki te \frac{7y}{10}+610.

$$x=-7y-6100$$

Whakakapia te -7y-6100 mō te x ki tērā atu whārite, -0.8x+0.5y+920=0.

$$-0.8\left(-7y-6100\right)+0.5y+920=0$$

Whakareatia -0.8 ki te -7y-6100.

$$5.6y+4880+0.5y+920=0$$

Tāpiri \frac{28y}{5} ki te \frac{y}{2}.

$$6.1y+4880+920=0$$

Tāpiri 4880 ki te 920.

$$6.1y+5800=0$$

Me tango 5800 mai i ngā taha e rua o te whārite.

$$6.1y=-5800$$

Whakawehea ngā taha e rua o te whārite ki te 6.1, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-\frac{58000}{61}$$

Whakaurua te -\frac{58000}{61} mō y ki x=-7y-6100. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-7\left(-\frac{58000}{61}\right)-6100$$

Whakareatia -7 ki te -\frac{58000}{61}.

$$x=\frac{406000}{61}-6100$$

Tāpiri -6100 ki te \frac{406000}{61}.

$$x=\frac{33900}{61}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{33900}{61},y=-\frac{58000}{61}$$

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.

$$-0.1x-0.7y-610=0,-0.8x+0.5y+920=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-0.1&-0.7\\-0.8&0.5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}610\\-920\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-0.1&-0.7\\-0.8&0.5\end{matrix}\right).

$$inverse(\left(\begin{matrix}-0.1&-0.7\\-0.8&0.5\end{matrix}\right))\left(\begin{matrix}-0.1&-0.7\\-0.8&0.5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-0.1&-0.7\\-0.8&0.5\end{matrix}\right))\left(\begin{matrix}610\\-920\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-0.1&-0.7\\-0.8&0.5\end{matrix}\right))\left(\begin{matrix}610\\-920\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-0.1&-0.7\\-0.8&0.5\end{matrix}\right))\left(\begin{matrix}610\\-920\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{0.5}{-0.1\times 0.5-\left(-0.7\left(-0.8\right)\right)}&-\frac{-0.7}{-0.1\times 0.5-\left(-0.7\left(-0.8\right)\right)}\\-\frac{-0.8}{-0.1\times 0.5-\left(-0.7\left(-0.8\right)\right)}&-\frac{0.1}{-0.1\times 0.5-\left(-0.7\left(-0.8\right)\right)}\end{matrix}\right)\left(\begin{matrix}610\\-920\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{50}{61}&-\frac{70}{61}\\-\frac{80}{61}&\frac{10}{61}\end{matrix}\right)\left(\begin{matrix}610\\-920\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{50}{61}\times 610-\frac{70}{61}\left(-920\right)\\-\frac{80}{61}\times 610+\frac{10}{61}\left(-920\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{33900}{61}\\-\frac{58000}{61}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{33900}{61},y=-\frac{58000}{61}$$

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.

$$-0.1x-0.7y-610=0,-0.8x+0.5y+920=0$$

Kia ōrite ai a -\frac{x}{10} me -\frac{4x}{5}, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -0.8 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te -0.1.

$$-0.8\left(-0.1\right)x-0.8\left(-0.7\right)y-0.8\left(-610\right)=0,-0.1\left(-0.8\right)x-0.1\times 0.5y-0.1\times 920=0$$

Whakarūnātia.

$$0.08x+0.56y+488=0,0.08x-0.05y-92=0$$

Me tango 0.08x-0.05y-92=0 mai i 0.08x+0.56y+488=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$0.08x-0.08x+0.56y+0.05y+488+92=0$$

Tāpiri \frac{2x}{25} ki te -\frac{2x}{25}. Ka whakakore atu ngā kupu \frac{2x}{25} me -\frac{2x}{25}, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$0.56y+0.05y+488+92=0$$

Tāpiri \frac{14y}{25} ki te \frac{y}{20}.

$$0.61y+488+92=0$$

Tāpiri 488 ki te 92.

$$0.61y+580=0$$

Me tango 580 mai i ngā taha e rua o te whārite.

$$0.61y=-580$$

Whakawehea ngā taha e rua o te whārite ki te 0.61, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-\frac{58000}{61}$$

Whakaurua te -\frac{58000}{61} mō y ki -0.8x+0.5y+920=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-0.8x+0.5\left(-\frac{58000}{61}\right)+920=0$$

Whakareatia 0.5 ki te -\frac{58000}{61} 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.

$$-0.8x-\frac{29000}{61}+920=0$$

Tāpiri -\frac{29000}{61} ki te 920.

$$-0.8x+\frac{27120}{61}=0$$

Me tango \frac{27120}{61} mai i ngā taha e rua o te whārite.

$$-0.8x=-\frac{27120}{61}$$

Whakawehea ngā taha e rua o te whārite ki te -0.8, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=\frac{33900}{61}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{33900}{61},y=-\frac{58000}{61}$$