Whakaoti i te {l}{0.4x+0.3y=1.7}{0.7x-0.2y=0.8}

$$\left. \begin{array} { l } { 0.4 x + 0.3 y = 1.7 } \\ { 0.7 x - 0.2 y = 0.8 } \end{array} \right.$$

Whakaoti mō x, y

$$x=2$$

$$y=3$$

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.

$$0.4x+0.3y=1.7,0.7x-0.2y=0.8$$

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

$$0.4x+0.3y=1.7$$

Me tango \frac{3y}{10} mai i ngā taha e rua o te whārite.

$$0.4x=-0.3y+1.7$$

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.

$$x=2.5\left(-0.3y+1.7\right)$$

Whakareatia 2.5 ki te \frac{-3y+17}{10}.

$$x=-0.75y+4.25$$

Whakakapia te \frac{-3y+17}{4} mō te x ki tērā atu whārite, 0.7x-0.2y=0.8.

$$0.7\left(-0.75y+4.25\right)-0.2y=0.8$$

Whakareatia 0.7 ki te \frac{-3y+17}{4}.

$$-0.525y+2.975-0.2y=0.8$$

Tāpiri -\frac{21y}{40} ki te -\frac{y}{5}.

$$-0.725y+2.975=0.8$$

Me tango 2.975 mai i ngā taha e rua o te whārite.

$$-0.725y=-2.175$$

Whakawehea ngā taha e rua o te whārite ki te -0.725, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=3$$

Whakaurua te 3 mō y ki x=-0.75y+4.25. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-0.75\times 3+4.25$$

Whakareatia -0.75 ki te 3.

$$x=\frac{-9+17}{4}$$

Tāpiri 4.25 ki te -2.25 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.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=3$$

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.

$$0.4x+0.3y=1.7,0.7x-0.2y=0.8$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}0.4&0.3\\0.7&-0.2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1.7\\0.8\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}0.4&0.3\\0.7&-0.2\end{matrix}\right).

$$inverse(\left(\begin{matrix}0.4&0.3\\0.7&-0.2\end{matrix}\right))\left(\begin{matrix}0.4&0.3\\0.7&-0.2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}0.4&0.3\\0.7&-0.2\end{matrix}\right))\left(\begin{matrix}1.7\\0.8\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}0.4&0.3\\0.7&-0.2\end{matrix}\right))\left(\begin{matrix}1.7\\0.8\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}0.4&0.3\\0.7&-0.2\end{matrix}\right))\left(\begin{matrix}1.7\\0.8\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{0.2}{0.4\left(-0.2\right)-0.3\times 0.7}&-\frac{0.3}{0.4\left(-0.2\right)-0.3\times 0.7}\\-\frac{0.7}{0.4\left(-0.2\right)-0.3\times 0.7}&\frac{0.4}{0.4\left(-0.2\right)-0.3\times 0.7}\end{matrix}\right)\left(\begin{matrix}1.7\\0.8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{20}{29}&\frac{30}{29}\\\frac{70}{29}&-\frac{40}{29}\end{matrix}\right)\left(\begin{matrix}1.7\\0.8\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{20}{29}\times 1.7+\frac{30}{29}\times 0.8\\\frac{70}{29}\times 1.7-\frac{40}{29}\times 0.8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2\\3\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=2,y=3$$

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.

$$0.4x+0.3y=1.7,0.7x-0.2y=0.8$$

Kia ōrite ai a \frac{2x}{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 0.4.

$$0.7\times 0.4x+0.7\times 0.3y=0.7\times 1.7,0.4\times 0.7x+0.4\left(-0.2\right)y=0.4\times 0.8$$

Whakarūnātia.

$$0.28x+0.21y=1.19,0.28x-0.08y=0.32$$

Me tango 0.28x-0.08y=0.32 mai i 0.28x+0.21y=1.19 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$0.28x-0.28x+0.21y+0.08y=1.19-0.32$$

Tāpiri \frac{7x}{25} ki te -\frac{7x}{25}. Ka whakakore atu ngā kupu \frac{7x}{25} me -\frac{7x}{25}, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$0.21y+0.08y=1.19-0.32$$

Tāpiri \frac{21y}{100} ki te \frac{2y}{25}.

$$0.29y=1.19-0.32$$

Tāpiri 1.19 ki te -0.32 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.

$$0.29y=0.87$$

Whakawehea ngā taha e rua o te whārite ki te 0.29, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=3$$

Whakaurua te 3 mō y ki 0.7x-0.2y=0.8. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$0.7x-0.2\times 3=0.8$$

Whakareatia -0.2 ki te 3.

$$0.7x-0.6=0.8$$

Me tāpiri 0.6 ki ngā taha e rua o te whārite.

$$0.7x=1.4$$

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.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=3$$