Whakaoti i te {l}{0.5x+y=9}{1.6x+0.2y=13}

$$\left. \begin{array} { l } { 0.5 x + y = 9 } \\ { 1.6 x + 0.2 y = 13 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{112}{15} = 7\frac{7}{15} \approx 7.466666667$$

$$y = \frac{79}{15} = 5\frac{4}{15} \approx 5.266666667$$

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.

$$0.5x+y=9,1.6x+0.2y=13$$

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

$$0.5x+y=9$$

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

$$0.5x=-y+9$$

Me whakarea ngā taha e rua ki te 2.

$$x=2\left(-y+9\right)$$

Whakareatia 2 ki te -y+9.

$$x=-2y+18$$

Whakakapia te -2y+18 mō te x ki tērā atu whārite, 1.6x+0.2y=13.

$$1.6\left(-2y+18\right)+0.2y=13$$

Whakareatia 1.6 ki te -2y+18.

$$-3.2y+28.8+0.2y=13$$

Tāpiri -\frac{16y}{5} ki te \frac{y}{5}.

$$-3y+28.8=13$$

Me tango 28.8 mai i ngā taha e rua o te whārite.

$$-3y=-15.8$$

Whakawehea ngā taha e rua ki te -3.

$$y=\frac{79}{15}$$

Whakaurua te \frac{79}{15} mō y ki x=-2y+18. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-2\times \frac{79}{15}+18$$

Whakareatia -2 ki te \frac{79}{15}.

$$x=-\frac{158}{15}+18$$

Tāpiri 18 ki te -\frac{158}{15}.

$$x=\frac{112}{15}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{112}{15},y=\frac{79}{15}$$

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.

$$0.5x+y=9,1.6x+0.2y=13$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}0.5&1\\1.6&0.2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}9\\13\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}0.5&1\\1.6&0.2\end{matrix}\right).

$$inverse(\left(\begin{matrix}0.5&1\\1.6&0.2\end{matrix}\right))\left(\begin{matrix}0.5&1\\1.6&0.2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}0.5&1\\1.6&0.2\end{matrix}\right))\left(\begin{matrix}9\\13\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}0.5&1\\1.6&0.2\end{matrix}\right))\left(\begin{matrix}9\\13\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}0.5&1\\1.6&0.2\end{matrix}\right))\left(\begin{matrix}9\\13\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{0.2}{0.5\times 0.2-1.6}&-\frac{1}{0.5\times 0.2-1.6}\\-\frac{1.6}{0.5\times 0.2-1.6}&\frac{0.5}{0.5\times 0.2-1.6}\end{matrix}\right)\left(\begin{matrix}9\\13\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{15}&\frac{2}{3}\\\frac{16}{15}&-\frac{1}{3}\end{matrix}\right)\left(\begin{matrix}9\\13\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{15}\times 9+\frac{2}{3}\times 13\\\frac{16}{15}\times 9-\frac{1}{3}\times 13\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{112}{15}\\\frac{79}{15}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{112}{15},y=\frac{79}{15}$$

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.

$$0.5x+y=9,1.6x+0.2y=13$$

Kia ōrite ai a \frac{x}{2} me \frac{8x}{5}, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 1.6 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 0.5.

$$1.6\times 0.5x+1.6y=1.6\times 9,0.5\times 1.6x+0.5\times 0.2y=0.5\times 13$$

Whakarūnātia.

$$0.8x+1.6y=14.4,0.8x+0.1y=6.5$$

Me tango 0.8x+0.1y=6.5 mai i 0.8x+1.6y=14.4 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$0.8x-0.8x+1.6y-0.1y=14.4-6.5$$

Tāpiri \frac{4x}{5} ki te -\frac{4x}{5}. Ka whakakore atu ngā kupu \frac{4x}{5} me -\frac{4x}{5}, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$1.6y-0.1y=14.4-6.5$$

Tāpiri \frac{8y}{5} ki te -\frac{y}{10}.

$$1.5y=14.4-6.5$$

Tāpiri 14.4 ki te -6.5 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.

$$1.5y=7.9$$

Whakawehea ngā taha e rua o te whārite ki te 1.5, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{79}{15}$$

Whakaurua te \frac{79}{15} mō y ki 1.6x+0.2y=13. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$1.6x+0.2\times \frac{79}{15}=13$$

Whakareatia 0.2 ki te \frac{79}{15} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$1.6x+\frac{79}{75}=13$$

Me tango \frac{79}{75} mai i ngā taha e rua o te whārite.

$$1.6x=\frac{896}{75}$$

Whakawehea ngā taha e rua o te whārite ki te 1.6, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=\frac{112}{15}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{112}{15},y=\frac{79}{15}$$