Whakaoti i te {l}{2(3x-4)+3y=31}{x/2-y/5=5}

$$\left\{ \begin{array} { l } { 2 ( 3 x - 4 ) + 3 y = 31 } \\ { \frac { x } { 2 } - \frac { y } { 5 } = 5 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{76}{9} = 8\frac{4}{9} \approx 8.444444444$$

$$y = -\frac{35}{9} = -3\frac{8}{9} \approx -3.888888889$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te 3x-4.

$$6x-8+3y=31$$

Me tāpiri te 8 ki ngā taha e rua.

$$6x+3y=31+8$$

Tāpirihia te 31 ki te 8, ka 39.

$$6x+3y=39$$

Whakaarohia te whārite tuarua. Me whakarea ngā taha e rua o te whārite ki te 10, arā, te tauraro pātahi he tino iti rawa te kitea o 2,5.

$$5x-2y=50$$

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.

$$6x+3y=39,5x-2y=50$$

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

$$6x+3y=39$$

Me tango 3y mai i ngā taha e rua o te whārite.

$$6x=-3y+39$$

Whakawehea ngā taha e rua ki te 6.

$$x=\frac{1}{6}\left(-3y+39\right)$$

Whakareatia \frac{1}{6} ki te -3y+39.

$$x=-\frac{1}{2}y+\frac{13}{2}$$

Whakakapia te \frac{-y+13}{2} mō te x ki tērā atu whārite, 5x-2y=50.

$$5\left(-\frac{1}{2}y+\frac{13}{2}\right)-2y=50$$

Whakareatia 5 ki te \frac{-y+13}{2}.

$$-\frac{5}{2}y+\frac{65}{2}-2y=50$$

Tāpiri -\frac{5y}{2} ki te -2y.

$$-\frac{9}{2}y+\frac{65}{2}=50$$

Me tango \frac{65}{2} mai i ngā taha e rua o te whārite.

$$-\frac{9}{2}y=\frac{35}{2}$$

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.

$$y=-\frac{35}{9}$$

Whakaurua te -\frac{35}{9} mō y ki x=-\frac{1}{2}y+\frac{13}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{2}\left(-\frac{35}{9}\right)+\frac{13}{2}$$

Whakareatia -\frac{1}{2} ki te -\frac{35}{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.

$$x=\frac{35}{18}+\frac{13}{2}$$

Tāpiri \frac{13}{2} ki te \frac{35}{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.

$$x=\frac{76}{9}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{76}{9},y=-\frac{35}{9}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te 3x-4.

$$6x-8+3y=31$$

Me tāpiri te 8 ki ngā taha e rua.

$$6x+3y=31+8$$

Tāpirihia te 31 ki te 8, ka 39.

$$6x+3y=39$$

Whakaarohia te whārite tuarua. Me whakarea ngā taha e rua o te whārite ki te 10, arā, te tauraro pātahi he tino iti rawa te kitea o 2,5.

$$5x-2y=50$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$6x+3y=39,5x-2y=50$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}6&3\\5&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}39\\50\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}6&3\\5&-2\end{matrix}\right).

$$inverse(\left(\begin{matrix}6&3\\5&-2\end{matrix}\right))\left(\begin{matrix}6&3\\5&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&3\\5&-2\end{matrix}\right))\left(\begin{matrix}39\\50\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&3\\5&-2\end{matrix}\right))\left(\begin{matrix}39\\50\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&3\\5&-2\end{matrix}\right))\left(\begin{matrix}39\\50\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{6\left(-2\right)-3\times 5}&-\frac{3}{6\left(-2\right)-3\times 5}\\-\frac{5}{6\left(-2\right)-3\times 5}&\frac{6}{6\left(-2\right)-3\times 5}\end{matrix}\right)\left(\begin{matrix}39\\50\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{27}&\frac{1}{9}\\\frac{5}{27}&-\frac{2}{9}\end{matrix}\right)\left(\begin{matrix}39\\50\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{27}\times 39+\frac{1}{9}\times 50\\\frac{5}{27}\times 39-\frac{2}{9}\times 50\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{76}{9}\\-\frac{35}{9}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{76}{9},y=-\frac{35}{9}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te 3x-4.

$$6x-8+3y=31$$

Me tāpiri te 8 ki ngā taha e rua.

$$6x+3y=31+8$$

Tāpirihia te 31 ki te 8, ka 39.

$$6x+3y=39$$

Whakaarohia te whārite tuarua. Me whakarea ngā taha e rua o te whārite ki te 10, arā, te tauraro pātahi he tino iti rawa te kitea o 2,5.

$$5x-2y=50$$

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.

$$6x+3y=39,5x-2y=50$$

Kia ōrite ai a 6x me 5x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 5 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 6.

$$5\times 6x+5\times 3y=5\times 39,6\times 5x+6\left(-2\right)y=6\times 50$$

Whakarūnātia.

$$30x+15y=195,30x-12y=300$$

Me tango 30x-12y=300 mai i 30x+15y=195 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$30x-30x+15y+12y=195-300$$

Tāpiri 30x ki te -30x. Ka whakakore atu ngā kupu 30x me -30x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$15y+12y=195-300$$

Tāpiri 15y ki te 12y.

$$27y=195-300$$

Tāpiri 195 ki te -300.

$$27y=-105$$

Whakawehea ngā taha e rua ki te 27.

$$y=-\frac{35}{9}$$

Whakaurua te -\frac{35}{9} mō y ki 5x-2y=50. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$5x-2\left(-\frac{35}{9}\right)=50$$

Whakareatia -2 ki te -\frac{35}{9}.

$$5x+\frac{70}{9}=50$$

Me tango \frac{70}{9} mai i ngā taha e rua o te whārite.

$$5x=\frac{380}{9}$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{76}{9}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{76}{9},y=-\frac{35}{9}$$