Whakaoti i te {l}{5x-2y=14}{3x+7y=21}

$$\left\{ \begin{array} { l } { 5 x - 2 y = 14 } \\ { 3 x + 7 y = 21 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{140}{41} = 3\frac{17}{41} \approx 3.414634146$$

$$y = \frac{63}{41} = 1\frac{22}{41} \approx 1.536585366$$

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.

$$5x-2y=14,3x+7y=21$$

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

$$5x-2y=14$$

Me tāpiri 2y ki ngā taha e rua o te whārite.

$$5x=2y+14$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\left(2y+14\right)$$

Whakareatia \frac{1}{5} ki te 14+2y.

$$x=\frac{2}{5}y+\frac{14}{5}$$

Whakakapia te \frac{14+2y}{5} mō te x ki tērā atu whārite, 3x+7y=21.

$$3\left(\frac{2}{5}y+\frac{14}{5}\right)+7y=21$$

Whakareatia 3 ki te \frac{14+2y}{5}.

$$\frac{6}{5}y+\frac{42}{5}+7y=21$$

Tāpiri \frac{6y}{5} ki te 7y.

$$\frac{41}{5}y+\frac{42}{5}=21$$

Me tango \frac{42}{5} mai i ngā taha e rua o te whārite.

$$\frac{41}{5}y=\frac{63}{5}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{41}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{63}{41}$$

Whakaurua te \frac{63}{41} mō y ki x=\frac{2}{5}y+\frac{14}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{2}{5}\times \frac{63}{41}+\frac{14}{5}$$

Whakareatia \frac{2}{5} ki te \frac{63}{41} 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.

$$x=\frac{126}{205}+\frac{14}{5}$$

Tāpiri \frac{14}{5} ki te \frac{126}{205} 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.

$$x=\frac{140}{41}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{140}{41},y=\frac{63}{41}$$

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.

$$5x-2y=14,3x+7y=21$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&-2\\3&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}14\\21\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&-2\\3&7\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&-2\\3&7\end{matrix}\right))\left(\begin{matrix}5&-2\\3&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-2\\3&7\end{matrix}\right))\left(\begin{matrix}14\\21\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-2\\3&7\end{matrix}\right))\left(\begin{matrix}14\\21\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-2\\3&7\end{matrix}\right))\left(\begin{matrix}14\\21\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{5\times 7-\left(-2\times 3\right)}&-\frac{-2}{5\times 7-\left(-2\times 3\right)}\\-\frac{3}{5\times 7-\left(-2\times 3\right)}&\frac{5}{5\times 7-\left(-2\times 3\right)}\end{matrix}\right)\left(\begin{matrix}14\\21\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{41}&\frac{2}{41}\\-\frac{3}{41}&\frac{5}{41}\end{matrix}\right)\left(\begin{matrix}14\\21\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{41}\times 14+\frac{2}{41}\times 21\\-\frac{3}{41}\times 14+\frac{5}{41}\times 21\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{140}{41}\\\frac{63}{41}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{140}{41},y=\frac{63}{41}$$

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.

$$5x-2y=14,3x+7y=21$$

Kia ōrite ai a 5x me 3x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 3 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 5.

$$3\times 5x+3\left(-2\right)y=3\times 14,5\times 3x+5\times 7y=5\times 21$$

Whakarūnātia.

$$15x-6y=42,15x+35y=105$$

Me tango 15x+35y=105 mai i 15x-6y=42 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$15x-15x-6y-35y=42-105$$

Tāpiri 15x ki te -15x. Ka whakakore atu ngā kupu 15x me -15x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-6y-35y=42-105$$

Tāpiri -6y ki te -35y.

$$-41y=42-105$$

Tāpiri 42 ki te -105.

$$-41y=-63$$

Whakawehea ngā taha e rua ki te -41.

$$y=\frac{63}{41}$$

Whakaurua te \frac{63}{41} mō y ki 3x+7y=21. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x+7\times \frac{63}{41}=21$$

Whakareatia 7 ki te \frac{63}{41}.

$$3x+\frac{441}{41}=21$$

Me tango \frac{441}{41} mai i ngā taha e rua o te whārite.

$$3x=\frac{420}{41}$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{140}{41}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{140}{41},y=\frac{63}{41}$$