Whakaoti i te {l}{2x+7y=5}{3x+6y=20}

$$\left. \begin{array} { l } { 2 x + 7 y = 5 } \\ { 3 x + 6 y = 20 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{110}{9} = 12\frac{2}{9} \approx 12.222222222$$

$$y = -\frac{25}{9} = -2\frac{7}{9} \approx -2.777777778$$

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.

$$2x+7y=5,3x+6y=20$$

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

$$2x+7y=5$$

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

$$2x=-7y+5$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(-7y+5\right)$$

Whakareatia \frac{1}{2} ki te -7y+5.

$$x=-\frac{7}{2}y+\frac{5}{2}$$

Whakakapia te \frac{-7y+5}{2} mō te x ki tērā atu whārite, 3x+6y=20.

$$3\left(-\frac{7}{2}y+\frac{5}{2}\right)+6y=20$$

Whakareatia 3 ki te \frac{-7y+5}{2}.

$$-\frac{21}{2}y+\frac{15}{2}+6y=20$$

Tāpiri -\frac{21y}{2} ki te 6y.

$$-\frac{9}{2}y+\frac{15}{2}=20$$

Me tango \frac{15}{2} mai i ngā taha e rua o te whārite.

$$-\frac{9}{2}y=\frac{25}{2}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{9}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-\frac{25}{9}$$

Whakaurua te -\frac{25}{9} mō y ki x=-\frac{7}{2}y+\frac{5}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{7}{2}\left(-\frac{25}{9}\right)+\frac{5}{2}$$

Whakareatia -\frac{7}{2} ki te -\frac{25}{9} 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.

$$x=\frac{175}{18}+\frac{5}{2}$$

Tāpiri \frac{5}{2} ki te \frac{175}{18} 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.

$$x=\frac{110}{9}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{110}{9},y=-\frac{25}{9}$$

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.

$$2x+7y=5,3x+6y=20$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&7\\3&6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\\20\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&7\\3&6\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&7\\3&6\end{matrix}\right))\left(\begin{matrix}2&7\\3&6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&7\\3&6\end{matrix}\right))\left(\begin{matrix}5\\20\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&7\\3&6\end{matrix}\right))\left(\begin{matrix}5\\20\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&7\\3&6\end{matrix}\right))\left(\begin{matrix}5\\20\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{6}{2\times 6-7\times 3}&-\frac{7}{2\times 6-7\times 3}\\-\frac{3}{2\times 6-7\times 3}&\frac{2}{2\times 6-7\times 3}\end{matrix}\right)\left(\begin{matrix}5\\20\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{3}&\frac{7}{9}\\\frac{1}{3}&-\frac{2}{9}\end{matrix}\right)\left(\begin{matrix}5\\20\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{3}\times 5+\frac{7}{9}\times 20\\\frac{1}{3}\times 5-\frac{2}{9}\times 20\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{110}{9}\\-\frac{25}{9}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{110}{9},y=-\frac{25}{9}$$

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.

$$2x+7y=5,3x+6y=20$$

Kia ōrite ai a 2x 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 2.

$$3\times 2x+3\times 7y=3\times 5,2\times 3x+2\times 6y=2\times 20$$

Whakarūnātia.

$$6x+21y=15,6x+12y=40$$

Me tango 6x+12y=40 mai i 6x+21y=15 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x+21y-12y=15-40$$

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.

$$21y-12y=15-40$$

Tāpiri 21y ki te -12y.

$$9y=15-40$$

Tāpiri 15 ki te -40.

$$9y=-25$$

Whakawehea ngā taha e rua ki te 9.

$$y=-\frac{25}{9}$$

Whakaurua te -\frac{25}{9} mō y ki 3x+6y=20. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x+6\left(-\frac{25}{9}\right)=20$$

Whakareatia 6 ki te -\frac{25}{9}.

$$3x-\frac{50}{3}=20$$

Me tāpiri \frac{50}{3} ki ngā taha e rua o te whārite.

$$3x=\frac{110}{3}$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{110}{9}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{110}{9},y=-\frac{25}{9}$$