Whakaoti i te {l}{3x+7y=63}{2x+4y=38}

$$\left. \begin{array} { l } { 3 x + 7 y = 63 } \\ { 2 x + 4 y = 38 } \end{array} \right.$$

Whakaoti mō x, y

$$x=7$$

$$y=6$$

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.

$$3x+7y=63,2x+4y=38$$

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

$$3x+7y=63$$

Me tango 7y mai i ngā taha e rua o te whārite.

$$3x=-7y+63$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(-7y+63\right)$$

Whakareatia \frac{1}{3} ki te -7y+63.

$$x=-\frac{7}{3}y+21$$

Whakakapia te -\frac{7y}{3}+21 mō te x ki tērā atu whārite, 2x+4y=38.

$$2\left(-\frac{7}{3}y+21\right)+4y=38$$

Whakareatia 2 ki te -\frac{7y}{3}+21.

$$-\frac{14}{3}y+42+4y=38$$

Tāpiri -\frac{14y}{3} ki te 4y.

$$-\frac{2}{3}y+42=38$$

Me tango 42 mai i ngā taha e rua o te whārite.

$$-\frac{2}{3}y=-4$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{2}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=6$$

Whakaurua te 6 mō y ki x=-\frac{7}{3}y+21. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{7}{3}\times 6+21$$

Whakareatia -\frac{7}{3} ki te 6.

$$x=-14+21$$

Tāpiri 21 ki te -14.

$$x=7$$

Kua oti te pūnaha te whakatau.

$$x=7,y=6$$

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.

$$3x+7y=63,2x+4y=38$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&7\\2&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}63\\38\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&7\\2&4\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&7\\2&4\end{matrix}\right))\left(\begin{matrix}3&7\\2&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&7\\2&4\end{matrix}\right))\left(\begin{matrix}63\\38\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&7\\2&4\end{matrix}\right))\left(\begin{matrix}63\\38\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&7\\2&4\end{matrix}\right))\left(\begin{matrix}63\\38\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te poukapa kōaro ko \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), kia tuhia anō ai te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{3\times 4-7\times 2}&-\frac{7}{3\times 4-7\times 2}\\-\frac{2}{3\times 4-7\times 2}&\frac{3}{3\times 4-7\times 2}\end{matrix}\right)\left(\begin{matrix}63\\38\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-2&\frac{7}{2}\\1&-\frac{3}{2}\end{matrix}\right)\left(\begin{matrix}63\\38\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-2\times 63+\frac{7}{2}\times 38\\63-\frac{3}{2}\times 38\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}7\\6\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=7,y=6$$

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.

$$3x+7y=63,2x+4y=38$$

Kia ōrite ai a 3x me 2x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$2\times 3x+2\times 7y=2\times 63,3\times 2x+3\times 4y=3\times 38$$

Whakarūnātia.

$$6x+14y=126,6x+12y=114$$

Me tango 6x+12y=114 mai i 6x+14y=126 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x+14y-12y=126-114$$

Tāpiri 6x ki te -6x. Ka whakakore atu ngā kupu 6x me -6x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$14y-12y=126-114$$

Tāpiri 14y ki te -12y.

$$2y=126-114$$

Tāpiri 126 ki te -114.

$$2y=12$$

Whakawehea ngā taha e rua ki te 2.

$$y=6$$

Whakaurua te 6 mō y ki 2x+4y=38. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x+4\times 6=38$$

Whakareatia 4 ki te 6.

$$2x+24=38$$

Me tango 24 mai i ngā taha e rua o te whārite.

$$2x=14$$

Whakawehea ngā taha e rua ki te 2.

$$x=7$$

Kua oti te pūnaha te whakatau.

$$x=7,y=6$$