Whakaoti i te {l}{0.4x+0.6y=-760}{-0.8x-0.3y=800}{text{Thesolutionis}}

$$\left. \begin{array} { l } { 0.4 x + 0.6 y = - 760 } \\ { - 0.8 x - 0.3 y = 800 } \\ { \text { The solution is } } \end{array} \right.$$

Whakaoti mō x, y

$$x=-700$$

$$y=-800$$

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.

$$0.4x+0.6y=-760,-0.8x-0.3y=800$$

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

$$0.4x+0.6y=-760$$

Me tango \frac{3y}{5} mai i ngā taha e rua o te whārite.

$$0.4x=-0.6y-760$$

Whakawehea ngā taha e rua o te whārite ki te 0.4, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=2.5\left(-0.6y-760\right)$$

Whakareatia 2.5 ki te -\frac{3y}{5}-760.

$$x=-1.5y-1900$$

Whakakapia te -\frac{3y}{2}-1900 mō te x ki tērā atu whārite, -0.8x-0.3y=800.

$$-0.8\left(-1.5y-1900\right)-0.3y=800$$

Whakareatia -0.8 ki te -\frac{3y}{2}-1900.

$$1.2y+1520-0.3y=800$$

Tāpiri \frac{6y}{5} ki te -\frac{3y}{10}.

$$0.9y+1520=800$$

Me tango 1520 mai i ngā taha e rua o te whārite.

$$0.9y=-720$$

Whakawehea ngā taha e rua o te whārite ki te 0.9, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-800$$

Whakaurua te -800 mō y ki x=-1.5y-1900. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-1.5\left(-800\right)-1900$$

Whakareatia -1.5 ki te -800.

$$x=1200-1900$$

Tāpiri -1900 ki te 1200.

$$x=-700$$

Kua oti te pūnaha te whakatau.

$$x=-700,y=-800$$

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.

$$0.4x+0.6y=-760,-0.8x-0.3y=800$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}0.4&0.6\\-0.8&-0.3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-760\\800\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}0.4&0.6\\-0.8&-0.3\end{matrix}\right).

$$inverse(\left(\begin{matrix}0.4&0.6\\-0.8&-0.3\end{matrix}\right))\left(\begin{matrix}0.4&0.6\\-0.8&-0.3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}0.4&0.6\\-0.8&-0.3\end{matrix}\right))\left(\begin{matrix}-760\\800\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}0.4&0.6\\-0.8&-0.3\end{matrix}\right))\left(\begin{matrix}-760\\800\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}0.4&0.6\\-0.8&-0.3\end{matrix}\right))\left(\begin{matrix}-760\\800\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{0.3}{0.4\left(-0.3\right)-0.6\left(-0.8\right)}&-\frac{0.6}{0.4\left(-0.3\right)-0.6\left(-0.8\right)}\\-\frac{-0.8}{0.4\left(-0.3\right)-0.6\left(-0.8\right)}&\frac{0.4}{0.4\left(-0.3\right)-0.6\left(-0.8\right)}\end{matrix}\right)\left(\begin{matrix}-760\\800\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{6}&-\frac{5}{3}\\\frac{20}{9}&\frac{10}{9}\end{matrix}\right)\left(\begin{matrix}-760\\800\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{6}\left(-760\right)-\frac{5}{3}\times 800\\\frac{20}{9}\left(-760\right)+\frac{10}{9}\times 800\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-700\\-800\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-700,y=-800$$

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.

$$0.4x+0.6y=-760,-0.8x-0.3y=800$$

Kia ōrite ai a \frac{2x}{5} 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 0.4.

$$-0.8\times 0.4x-0.8\times 0.6y=-0.8\left(-760\right),0.4\left(-0.8\right)x+0.4\left(-0.3\right)y=0.4\times 800$$

Whakarūnātia.

$$-0.32x-0.48y=608,-0.32x-0.12y=320$$

Me tango -0.32x-0.12y=320 mai i -0.32x-0.48y=608 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-0.32x+0.32x-0.48y+0.12y=608-320$$

Tāpiri -\frac{8x}{25} ki te \frac{8x}{25}. Ka whakakore atu ngā kupu -\frac{8x}{25} me \frac{8x}{25}, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-0.48y+0.12y=608-320$$

Tāpiri -\frac{12y}{25} ki te \frac{3y}{25}.

$$-0.36y=608-320$$

Tāpiri 608 ki te -320.

$$-0.36y=288$$

Whakawehea ngā taha e rua o te whārite ki te -0.36, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-800$$

Whakaurua te -800 mō y ki -0.8x-0.3y=800. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-0.8x-0.3\left(-800\right)=800$$

Whakareatia -0.3 ki te -800.

$$-0.8x+240=800$$

Me tango 240 mai i ngā taha e rua o te whārite.

$$-0.8x=560$$

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.

$$x=-700$$

Kua oti te pūnaha te whakatau.

$$x=-700,y=-800$$