Whakaoti i te {l}{-0.5x+0.1y=350}{0.4x+0.2y=0}{text{Thesolutionis}}

$$\left. \begin{array} { l } { - 0.5 x + 0.1 y = 350 } \\ { 0.4 x + 0.2 y = 0 } \\ { \text { The solution is } } \end{array} \right.$$

Whakaoti mō x, y

$$x=-500$$

$$y=1000$$

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.

$$-0.5x+0.1y=350,0.4x+0.2y=0$$

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

$$-0.5x+0.1y=350$$

Me tango \frac{y}{10} mai i ngā taha e rua o te whārite.

$$-0.5x=-0.1y+350$$

Me whakarea ngā taha e rua ki te -2.

$$x=-2\left(-0.1y+350\right)$$

Whakareatia -2 ki te -\frac{y}{10}+350.

$$x=0.2y-700$$

Whakakapia te \frac{y}{5}-700 mō te x ki tērā atu whārite, 0.4x+0.2y=0.

$$0.4\left(0.2y-700\right)+0.2y=0$$

Whakareatia 0.4 ki te \frac{y}{5}-700.

$$0.08y-280+0.2y=0$$

Tāpiri \frac{2y}{25} ki te \frac{y}{5}.

$$0.28y-280=0$$

Me tāpiri 280 ki ngā taha e rua o te whārite.

$$0.28y=280$$

Whakawehea ngā taha e rua o te whārite ki te 0.28, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=1000$$

Whakaurua te 1000 mō y ki x=0.2y-700. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=0.2\times 1000-700$$

Whakareatia 0.2 ki te 1000.

$$x=200-700$$

Tāpiri -700 ki te 200.

$$x=-500$$

Kua oti te pūnaha te whakatau.

$$x=-500,y=1000$$

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.

$$-0.5x+0.1y=350,0.4x+0.2y=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-0.5&0.1\\0.4&0.2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}350\\0\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-0.5&0.1\\0.4&0.2\end{matrix}\right).

$$inverse(\left(\begin{matrix}-0.5&0.1\\0.4&0.2\end{matrix}\right))\left(\begin{matrix}-0.5&0.1\\0.4&0.2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-0.5&0.1\\0.4&0.2\end{matrix}\right))\left(\begin{matrix}350\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-0.5&0.1\\0.4&0.2\end{matrix}\right))\left(\begin{matrix}350\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-0.5&0.1\\0.4&0.2\end{matrix}\right))\left(\begin{matrix}350\\0\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te poukapa kōaro ko \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), kia tuhia anō ai te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{0.2}{-0.5\times 0.2-0.1\times 0.4}&-\frac{0.1}{-0.5\times 0.2-0.1\times 0.4}\\-\frac{0.4}{-0.5\times 0.2-0.1\times 0.4}&-\frac{0.5}{-0.5\times 0.2-0.1\times 0.4}\end{matrix}\right)\left(\begin{matrix}350\\0\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{10}{7}&\frac{5}{7}\\\frac{20}{7}&\frac{25}{7}\end{matrix}\right)\left(\begin{matrix}350\\0\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{10}{7}\times 350\\\frac{20}{7}\times 350\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-500\\1000\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-500,y=1000$$

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.

$$-0.5x+0.1y=350,0.4x+0.2y=0$$

Kia ōrite ai a -\frac{x}{2} me \frac{2x}{5}, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 0.4 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te -0.5.

$$0.4\left(-0.5\right)x+0.4\times 0.1y=0.4\times 350,-0.5\times 0.4x-0.5\times 0.2y=0$$

Whakarūnātia.

$$-0.2x+0.04y=140,-0.2x-0.1y=0$$

Me tango -0.2x-0.1y=0 mai i -0.2x+0.04y=140 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-0.2x+0.2x+0.04y+0.1y=140$$

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

$$0.04y+0.1y=140$$

Tāpiri \frac{y}{25} ki te \frac{y}{10}.

$$0.14y=140$$

Whakawehea ngā taha e rua o te whārite ki te 0.14, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=1000$$

Whakaurua te 1000 mō y ki 0.4x+0.2y=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$0.4x+0.2\times 1000=0$$

Whakareatia 0.2 ki te 1000.

$$0.4x+200=0$$

Me tango 200 mai i ngā taha e rua o te whārite.

$$0.4x=-200$$

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.

$$x=-500$$

Kua oti te pūnaha te whakatau.

$$x=-500,y=1000$$