Whakaoti i te {l}{80x+160y=4}{x+3y=0.1}

$$\left. \begin{array} { l } { 80 x + 160 y = 4 } \\ { x + 3 y = 0.1 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-0.05$$

$$y=0.05$$

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.

$$80x+160y=4,x+3y=0.1$$

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

$$80x+160y=4$$

Me tango 160y mai i ngā taha e rua o te whārite.

$$80x=-160y+4$$

Whakawehea ngā taha e rua ki te 80.

$$x=\frac{1}{80}\left(-160y+4\right)$$

Whakareatia \frac{1}{80} ki te -160y+4.

$$x=-2y+\frac{1}{20}$$

Whakakapia te -2y+\frac{1}{20} mō te x ki tērā atu whārite, x+3y=0.1.

$$-2y+\frac{1}{20}+3y=0.1$$

Tāpiri -2y ki te 3y.

$$y+\frac{1}{20}=0.1$$

Me tango \frac{1}{20} mai i ngā taha e rua o te whārite.

$$y=\frac{1}{20}$$

Whakaurua te \frac{1}{20} mō y ki x=-2y+\frac{1}{20}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-2\times \frac{1}{20}+\frac{1}{20}$$

Whakareatia -2 ki te \frac{1}{20}.

$$x=-\frac{1}{10}+\frac{1}{20}$$

Tāpiri \frac{1}{20} ki te -\frac{1}{10} 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.

$$x=-\frac{1}{20}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{1}{20},y=\frac{1}{20}$$

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.

$$80x+160y=4,x+3y=0.1$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}80&160\\1&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\\0.1\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}80&160\\1&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}80&160\\1&3\end{matrix}\right))\left(\begin{matrix}80&160\\1&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}80&160\\1&3\end{matrix}\right))\left(\begin{matrix}4\\0.1\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}80&160\\1&3\end{matrix}\right))\left(\begin{matrix}4\\0.1\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}80&160\\1&3\end{matrix}\right))\left(\begin{matrix}4\\0.1\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{80\times 3-160}&-\frac{160}{80\times 3-160}\\-\frac{1}{80\times 3-160}&\frac{80}{80\times 3-160}\end{matrix}\right)\left(\begin{matrix}4\\0.1\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{80}&-2\\-\frac{1}{80}&1\end{matrix}\right)\left(\begin{matrix}4\\0.1\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{80}\times 4-2\times 0.1\\-\frac{1}{80}\times 4+0.1\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{20}\\\frac{1}{20}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{1}{20},y=\frac{1}{20}$$

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.

$$80x+160y=4,x+3y=0.1$$

Kia ōrite ai a 80x me x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 1 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 80.

$$80x+160y=4,80x+80\times 3y=80\times 0.1$$

Whakarūnātia.

$$80x+160y=4,80x+240y=8$$

Me tango 80x+240y=8 mai i 80x+160y=4 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$80x-80x+160y-240y=4-8$$

Tāpiri 80x ki te -80x. Ka whakakore atu ngā kupu 80x me -80x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$160y-240y=4-8$$

Tāpiri 160y ki te -240y.

$$-80y=4-8$$

Tāpiri 4 ki te -8.

$$-80y=-4$$

Whakawehea ngā taha e rua ki te -80.

$$y=\frac{1}{20}$$

Whakaurua te \frac{1}{20} mō y ki x+3y=0.1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x+3\times \frac{1}{20}=0.1$$

Whakareatia 3 ki te \frac{1}{20}.

$$x+\frac{3}{20}=0.1$$

Me tango \frac{3}{20} mai i ngā taha e rua o te whārite.

$$x=-\frac{1}{20}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{1}{20},y=\frac{1}{20}$$