Whakaoti i te {l}{0.3x-0.5y=29}{0.9x=0.2y+19}

$$\left\{ \begin{array} { l } { 0.3 x - 0.5 y = 29 } \\ { 0.9 x = 0.2 y + 19 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{370}{39} = 9\frac{19}{39} \approx 9.487179487$$

$$y = -\frac{680}{13} = -52\frac{4}{13} \approx -52.307692308$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Tangohia te 0.2y mai i ngā taha e rua.

$$0.9x-0.2y=19$$

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.

$$0.3x-0.5y=29,0.9x-0.2y=19$$

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

$$0.3x-0.5y=29$$

Me tāpiri \frac{y}{2} ki ngā taha e rua o te whārite.

$$0.3x=0.5y+29$$

Whakawehea ngā taha e rua o te whārite ki te 0.3, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=\frac{10}{3}\left(0.5y+29\right)$$

Whakareatia \frac{10}{3} ki te \frac{y}{2}+29.

$$x=\frac{5}{3}y+\frac{290}{3}$$

Whakakapia te \frac{5y+290}{3} mō te x ki tērā atu whārite, 0.9x-0.2y=19.

$$0.9\left(\frac{5}{3}y+\frac{290}{3}\right)-0.2y=19$$

Whakareatia 0.9 ki te \frac{5y+290}{3}.

$$1.5y+87-0.2y=19$$

Tāpiri \frac{3y}{2} ki te -\frac{y}{5}.

$$1.3y+87=19$$

Me tango 87 mai i ngā taha e rua o te whārite.

$$1.3y=-68$$

Whakawehea ngā taha e rua o te whārite ki te 1.3, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-\frac{680}{13}$$

Whakaurua te -\frac{680}{13} mō y ki x=\frac{5}{3}y+\frac{290}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{5}{3}\left(-\frac{680}{13}\right)+\frac{290}{3}$$

Whakareatia \frac{5}{3} ki te -\frac{680}{13} 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.

$$x=-\frac{3400}{39}+\frac{290}{3}$$

Tāpiri \frac{290}{3} ki te -\frac{3400}{39} 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.

$$x=\frac{370}{39}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{370}{39},y=-\frac{680}{13}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Tangohia te 0.2y mai i ngā taha e rua.

$$0.9x-0.2y=19$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$0.3x-0.5y=29,0.9x-0.2y=19$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}0.3&-0.5\\0.9&-0.2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}29\\19\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}0.3&-0.5\\0.9&-0.2\end{matrix}\right).

$$inverse(\left(\begin{matrix}0.3&-0.5\\0.9&-0.2\end{matrix}\right))\left(\begin{matrix}0.3&-0.5\\0.9&-0.2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}0.3&-0.5\\0.9&-0.2\end{matrix}\right))\left(\begin{matrix}29\\19\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}0.3&-0.5\\0.9&-0.2\end{matrix}\right))\left(\begin{matrix}29\\19\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}0.3&-0.5\\0.9&-0.2\end{matrix}\right))\left(\begin{matrix}29\\19\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{0.2}{0.3\left(-0.2\right)-\left(-0.5\times 0.9\right)}&-\frac{-0.5}{0.3\left(-0.2\right)-\left(-0.5\times 0.9\right)}\\-\frac{0.9}{0.3\left(-0.2\right)-\left(-0.5\times 0.9\right)}&\frac{0.3}{0.3\left(-0.2\right)-\left(-0.5\times 0.9\right)}\end{matrix}\right)\left(\begin{matrix}29\\19\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{20}{39}&\frac{50}{39}\\-\frac{30}{13}&\frac{10}{13}\end{matrix}\right)\left(\begin{matrix}29\\19\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{20}{39}\times 29+\frac{50}{39}\times 19\\-\frac{30}{13}\times 29+\frac{10}{13}\times 19\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{370}{39}\\-\frac{680}{13}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{370}{39},y=-\frac{680}{13}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Tangohia te 0.2y mai i ngā taha e rua.

$$0.9x-0.2y=19$$

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.

$$0.3x-0.5y=29,0.9x-0.2y=19$$

Kia ōrite ai a \frac{3x}{10} me \frac{9x}{10}, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 0.9 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 0.3.

$$0.9\times 0.3x+0.9\left(-0.5\right)y=0.9\times 29,0.3\times 0.9x+0.3\left(-0.2\right)y=0.3\times 19$$

Whakarūnātia.

$$0.27x-0.45y=26.1,0.27x-0.06y=5.7$$

Me tango 0.27x-0.06y=5.7 mai i 0.27x-0.45y=26.1 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$0.27x-0.27x-0.45y+0.06y=\frac{261-57}{10}$$

Tāpiri \frac{27x}{100} ki te -\frac{27x}{100}. Ka whakakore atu ngā kupu \frac{27x}{100} me -\frac{27x}{100}, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-0.45y+0.06y=\frac{261-57}{10}$$

Tāpiri -\frac{9y}{20} ki te \frac{3y}{50}.

$$-0.39y=\frac{261-57}{10}$$

Tāpiri 26.1 ki te -5.7 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.

$$-0.39y=20.4$$

Whakawehea ngā taha e rua o te whārite ki te -0.39, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-\frac{680}{13}$$

Whakaurua te -\frac{680}{13} mō y ki 0.9x-0.2y=19. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$0.9x-0.2\left(-\frac{680}{13}\right)=19$$

Whakareatia -0.2 ki te -\frac{680}{13} 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.

$$0.9x+\frac{136}{13}=19$$

Me tango \frac{136}{13} mai i ngā taha e rua o te whārite.

$$0.9x=\frac{111}{13}$$

Whakawehea ngā taha e rua o te whārite ki te 0.9, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=\frac{370}{39}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{370}{39},y=-\frac{680}{13}$$