Whakaoti i te {l}{5x-4y=19}{3x+2y=71}

$$\left\{ \begin{array} { l } { 5 x - 4 y = 19 } \\ { 3 x + 2 y = 71 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{161}{11} = 14\frac{7}{11} \approx 14.636363636$$

$$y = \frac{149}{11} = 13\frac{6}{11} \approx 13.545454545$$

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.

$$5x-4y=19,3x+2y=71$$

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

$$5x-4y=19$$

Me tāpiri 4y ki ngā taha e rua o te whārite.

$$5x=4y+19$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\left(4y+19\right)$$

Whakareatia \frac{1}{5} ki te 4y+19.

$$x=\frac{4}{5}y+\frac{19}{5}$$

Whakakapia te \frac{4y+19}{5} mō te x ki tērā atu whārite, 3x+2y=71.

$$3\left(\frac{4}{5}y+\frac{19}{5}\right)+2y=71$$

Whakareatia 3 ki te \frac{4y+19}{5}.

$$\frac{12}{5}y+\frac{57}{5}+2y=71$$

Tāpiri \frac{12y}{5} ki te 2y.

$$\frac{22}{5}y+\frac{57}{5}=71$$

Me tango \frac{57}{5} mai i ngā taha e rua o te whārite.

$$\frac{22}{5}y=\frac{298}{5}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{22}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{149}{11}$$

Whakaurua te \frac{149}{11} mō y ki x=\frac{4}{5}y+\frac{19}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{4}{5}\times \frac{149}{11}+\frac{19}{5}$$

Whakareatia \frac{4}{5} ki te \frac{149}{11} 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.

$$x=\frac{596}{55}+\frac{19}{5}$$

Tāpiri \frac{19}{5} ki te \frac{596}{55} 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.

$$x=\frac{161}{11}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{161}{11},y=\frac{149}{11}$$

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.

$$5x-4y=19,3x+2y=71$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&-4\\3&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}19\\71\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&-4\\3&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&-4\\3&2\end{matrix}\right))\left(\begin{matrix}5&-4\\3&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-4\\3&2\end{matrix}\right))\left(\begin{matrix}19\\71\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-4\\3&2\end{matrix}\right))\left(\begin{matrix}19\\71\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-4\\3&2\end{matrix}\right))\left(\begin{matrix}19\\71\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{5\times 2-\left(-4\times 3\right)}&-\frac{-4}{5\times 2-\left(-4\times 3\right)}\\-\frac{3}{5\times 2-\left(-4\times 3\right)}&\frac{5}{5\times 2-\left(-4\times 3\right)}\end{matrix}\right)\left(\begin{matrix}19\\71\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{11}&\frac{2}{11}\\-\frac{3}{22}&\frac{5}{22}\end{matrix}\right)\left(\begin{matrix}19\\71\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{11}\times 19+\frac{2}{11}\times 71\\-\frac{3}{22}\times 19+\frac{5}{22}\times 71\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{161}{11}\\\frac{149}{11}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{161}{11},y=\frac{149}{11}$$

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.

$$5x-4y=19,3x+2y=71$$

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.

$$3\times 5x+3\left(-4\right)y=3\times 19,5\times 3x+5\times 2y=5\times 71$$

Whakarūnātia.

$$15x-12y=57,15x+10y=355$$

Me tango 15x+10y=355 mai i 15x-12y=57 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$15x-15x-12y-10y=57-355$$

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.

$$-12y-10y=57-355$$

Tāpiri -12y ki te -10y.

$$-22y=57-355$$

Tāpiri 57 ki te -355.

$$-22y=-298$$

Whakawehea ngā taha e rua ki te -22.

$$y=\frac{149}{11}$$

Whakaurua te \frac{149}{11} mō y ki 3x+2y=71. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x+2\times \frac{149}{11}=71$$

Whakareatia 2 ki te \frac{149}{11}.

$$3x+\frac{298}{11}=71$$

Me tango \frac{298}{11} mai i ngā taha e rua o te whārite.

$$3x=\frac{483}{11}$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{161}{11}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{161}{11},y=\frac{149}{11}$$