Whakaoti i te {l}{3x+6y=24}{9x+5y=68}

$$\left\{ \begin{array} { l } { 3 x + 6 y = 24 } \\ { 9 x + 5 y = 68 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{96}{13} = 7\frac{5}{13} \approx 7.384615385$$

$$y=\frac{4}{13}\approx 0.307692308$$

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.

$$3x+6y=24,9x+5y=68$$

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

$$3x+6y=24$$

Me tango 6y mai i ngā taha e rua o te whārite.

$$3x=-6y+24$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(-6y+24\right)$$

Whakareatia \frac{1}{3} ki te -6y+24.

$$x=-2y+8$$

Whakakapia te -2y+8 mō te x ki tērā atu whārite, 9x+5y=68.

$$9\left(-2y+8\right)+5y=68$$

Whakareatia 9 ki te -2y+8.

$$-18y+72+5y=68$$

Tāpiri -18y ki te 5y.

$$-13y+72=68$$

Me tango 72 mai i ngā taha e rua o te whārite.

$$-13y=-4$$

Whakawehea ngā taha e rua ki te -13.

$$y=\frac{4}{13}$$

Whakaurua te \frac{4}{13} mō y ki x=-2y+8. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-2\times \frac{4}{13}+8$$

Whakareatia -2 ki te \frac{4}{13}.

$$x=-\frac{8}{13}+8$$

Tāpiri 8 ki te -\frac{8}{13}.

$$x=\frac{96}{13}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{96}{13},y=\frac{4}{13}$$

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.

$$3x+6y=24,9x+5y=68$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&6\\9&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}24\\68\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&6\\9&5\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&6\\9&5\end{matrix}\right))\left(\begin{matrix}3&6\\9&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&6\\9&5\end{matrix}\right))\left(\begin{matrix}24\\68\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&6\\9&5\end{matrix}\right))\left(\begin{matrix}24\\68\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&6\\9&5\end{matrix}\right))\left(\begin{matrix}24\\68\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{3\times 5-6\times 9}&-\frac{6}{3\times 5-6\times 9}\\-\frac{9}{3\times 5-6\times 9}&\frac{3}{3\times 5-6\times 9}\end{matrix}\right)\left(\begin{matrix}24\\68\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{39}&\frac{2}{13}\\\frac{3}{13}&-\frac{1}{13}\end{matrix}\right)\left(\begin{matrix}24\\68\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{39}\times 24+\frac{2}{13}\times 68\\\frac{3}{13}\times 24-\frac{1}{13}\times 68\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{96}{13}\\\frac{4}{13}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{96}{13},y=\frac{4}{13}$$

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.

$$3x+6y=24,9x+5y=68$$

Kia ōrite ai a 3x me 9x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 9 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$9\times 3x+9\times 6y=9\times 24,3\times 9x+3\times 5y=3\times 68$$

Whakarūnātia.

$$27x+54y=216,27x+15y=204$$

Me tango 27x+15y=204 mai i 27x+54y=216 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$27x-27x+54y-15y=216-204$$

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

$$54y-15y=216-204$$

Tāpiri 54y ki te -15y.

$$39y=216-204$$

Tāpiri 216 ki te -204.

$$39y=12$$

Whakawehea ngā taha e rua ki te 39.

$$y=\frac{4}{13}$$

Whakaurua te \frac{4}{13} mō y ki 9x+5y=68. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$9x+5\times \frac{4}{13}=68$$

Whakareatia 5 ki te \frac{4}{13}.

$$9x+\frac{20}{13}=68$$

Me tango \frac{20}{13} mai i ngā taha e rua o te whārite.

$$9x=\frac{864}{13}$$

Whakawehea ngā taha e rua ki te 9.

$$x=\frac{96}{13}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{96}{13},y=\frac{4}{13}$$