Whakaoti i te {l}{0.2x+0.1y=-180}{-0.7x-0.2y=480}{text{Thesolutionis}}

$$\left. \begin{array} { l } { 0.2 x + 0.1 y = - 180 } \\ { - 0.7 x - 0.2 y = 480 } \\ { \text { The solution is } } \end{array} \right.$$

Whakaoti mō x, y

$$x=-400$$

$$y=-1000$$

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.

$$0.2x+0.1y=-180,-0.7x-0.2y=480$$

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

$$0.2x+0.1y=-180$$

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

$$0.2x=-0.1y-180$$

Me whakarea ngā taha e rua ki te 5.

$$x=5\left(-0.1y-180\right)$$

Whakareatia 5 ki te -\frac{y}{10}-180.

$$x=-0.5y-900$$

Whakakapia te -\frac{y}{2}-900 mō te x ki tērā atu whārite, -0.7x-0.2y=480.

$$-0.7\left(-0.5y-900\right)-0.2y=480$$

Whakareatia -0.7 ki te -\frac{y}{2}-900.

$$0.35y+630-0.2y=480$$

Tāpiri \frac{7y}{20} ki te -\frac{y}{5}.

$$0.15y+630=480$$

Me tango 630 mai i ngā taha e rua o te whārite.

$$0.15y=-150$$

Whakawehea ngā taha e rua o te whārite ki te 0.15, 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=-0.5y-900. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-0.5\left(-1000\right)-900$$

Whakareatia -0.5 ki te -1000.

$$x=500-900$$

Tāpiri -900 ki te 500.

$$x=-400$$

Kua oti te pūnaha te whakatau.

$$x=-400,y=-1000$$

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.

$$0.2x+0.1y=-180,-0.7x-0.2y=480$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}0.2&0.1\\-0.7&-0.2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-180\\480\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}0.2&0.1\\-0.7&-0.2\end{matrix}\right).

$$inverse(\left(\begin{matrix}0.2&0.1\\-0.7&-0.2\end{matrix}\right))\left(\begin{matrix}0.2&0.1\\-0.7&-0.2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}0.2&0.1\\-0.7&-0.2\end{matrix}\right))\left(\begin{matrix}-180\\480\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}0.2&0.1\\-0.7&-0.2\end{matrix}\right))\left(\begin{matrix}-180\\480\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}0.2&0.1\\-0.7&-0.2\end{matrix}\right))\left(\begin{matrix}-180\\480\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{0.2}{0.2\left(-0.2\right)-0.1\left(-0.7\right)}&-\frac{0.1}{0.2\left(-0.2\right)-0.1\left(-0.7\right)}\\-\frac{-0.7}{0.2\left(-0.2\right)-0.1\left(-0.7\right)}&\frac{0.2}{0.2\left(-0.2\right)-0.1\left(-0.7\right)}\end{matrix}\right)\left(\begin{matrix}-180\\480\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{20}{3}&-\frac{10}{3}\\\frac{70}{3}&\frac{20}{3}\end{matrix}\right)\left(\begin{matrix}-180\\480\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{20}{3}\left(-180\right)-\frac{10}{3}\times 480\\\frac{70}{3}\left(-180\right)+\frac{20}{3}\times 480\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-400\\-1000\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-400,y=-1000$$

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.

$$0.2x+0.1y=-180,-0.7x-0.2y=480$$

Kia ōrite ai a \frac{x}{5} me -\frac{7x}{10}, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -0.7 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 0.2.

$$-0.7\times 0.2x-0.7\times 0.1y=-0.7\left(-180\right),0.2\left(-0.7\right)x+0.2\left(-0.2\right)y=0.2\times 480$$

Whakarūnātia.

$$-0.14x-0.07y=126,-0.14x-0.04y=96$$

Me tango -0.14x-0.04y=96 mai i -0.14x-0.07y=126 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-0.14x+0.14x-0.07y+0.04y=126-96$$

Tāpiri -\frac{7x}{50} ki te \frac{7x}{50}. Ka whakakore atu ngā kupu -\frac{7x}{50} me \frac{7x}{50}, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-0.07y+0.04y=126-96$$

Tāpiri -\frac{7y}{100} ki te \frac{y}{25}.

$$-0.03y=126-96$$

Tāpiri 126 ki te -96.

$$-0.03y=30$$

Whakawehea ngā taha e rua o te whārite ki te -0.03, 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 -0.7x-0.2y=480. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-0.7x-0.2\left(-1000\right)=480$$

Whakareatia -0.2 ki te -1000.

$$-0.7x+200=480$$

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

$$-0.7x=280$$

Whakawehea ngā taha e rua o te whārite ki te -0.7, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=-400$$

Kua oti te pūnaha te whakatau.

$$x=-400,y=-1000$$