Whakaoti i te {l}{9x+4y/3-5x-11/2=13-y}{13x-7y=-8}

$$\left. \begin{array} { l } { \frac { 9 x + 4 y } { 3 } - \frac { 5 x - 11 } { 2 } = 13 - y } \\ { 13 x - 7 y = - 8 } \end{array} \right.$$

Whakaoti mō x, y

$$x=1$$

$$y=3$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua o te whārite ki te 6, arā, te tauraro pātahi he tino iti rawa te kitea o 3,2.

$$2\left(9x+4y\right)-3\left(5x-11\right)=78-6y$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te 9x+4y.

$$18x+8y-3\left(5x-11\right)=78-6y$$

Whakamahia te āhuatanga tohatoha hei whakarea te -3 ki te 5x-11.

$$18x+8y-15x+33=78-6y$$

Pahekotia te 18x me -15x, ka 3x.

$$3x+8y+33=78-6y$$

Me tāpiri te 6y ki ngā taha e rua.

$$3x+8y+33+6y=78$$

Pahekotia te 8y me 6y, ka 14y.

$$3x+14y+33=78$$

Tangohia te 33 mai i ngā taha e rua.

$$3x+14y=78-33$$

Tangohia te 33 i te 78, ka 45.

$$3x+14y=45$$

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.

$$3x+14y=45,13x-7y=-8$$

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

$$3x+14y=45$$

Me tango 14y mai i ngā taha e rua o te whārite.

$$3x=-14y+45$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(-14y+45\right)$$

Whakareatia \frac{1}{3} ki te -14y+45.

$$x=-\frac{14}{3}y+15$$

Whakakapia te -\frac{14y}{3}+15 mō te x ki tērā atu whārite, 13x-7y=-8.

$$13\left(-\frac{14}{3}y+15\right)-7y=-8$$

Whakareatia 13 ki te -\frac{14y}{3}+15.

$$-\frac{182}{3}y+195-7y=-8$$

Tāpiri -\frac{182y}{3} ki te -7y.

$$-\frac{203}{3}y+195=-8$$

Me tango 195 mai i ngā taha e rua o te whārite.

$$-\frac{203}{3}y=-203$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{203}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=3$$

Whakaurua te 3 mō y ki x=-\frac{14}{3}y+15. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{14}{3}\times 3+15$$

Whakareatia -\frac{14}{3} ki te 3.

$$x=-14+15$$

Tāpiri 15 ki te -14.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=3$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua o te whārite ki te 6, arā, te tauraro pātahi he tino iti rawa te kitea o 3,2.

$$2\left(9x+4y\right)-3\left(5x-11\right)=78-6y$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te 9x+4y.

$$18x+8y-3\left(5x-11\right)=78-6y$$

Whakamahia te āhuatanga tohatoha hei whakarea te -3 ki te 5x-11.

$$18x+8y-15x+33=78-6y$$

Pahekotia te 18x me -15x, ka 3x.

$$3x+8y+33=78-6y$$

Me tāpiri te 6y ki ngā taha e rua.

$$3x+8y+33+6y=78$$

Pahekotia te 8y me 6y, ka 14y.

$$3x+14y+33=78$$

Tangohia te 33 mai i ngā taha e rua.

$$3x+14y=78-33$$

Tangohia te 33 i te 78, ka 45.

$$3x+14y=45$$

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

$$3x+14y=45,13x-7y=-8$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&14\\13&-7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}45\\-8\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&14\\13&-7\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&14\\13&-7\end{matrix}\right))\left(\begin{matrix}3&14\\13&-7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&14\\13&-7\end{matrix}\right))\left(\begin{matrix}45\\-8\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&14\\13&-7\end{matrix}\right))\left(\begin{matrix}45\\-8\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&14\\13&-7\end{matrix}\right))\left(\begin{matrix}45\\-8\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{7}{3\left(-7\right)-14\times 13}&-\frac{14}{3\left(-7\right)-14\times 13}\\-\frac{13}{3\left(-7\right)-14\times 13}&\frac{3}{3\left(-7\right)-14\times 13}\end{matrix}\right)\left(\begin{matrix}45\\-8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{29}&\frac{2}{29}\\\frac{13}{203}&-\frac{3}{203}\end{matrix}\right)\left(\begin{matrix}45\\-8\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{29}\times 45+\frac{2}{29}\left(-8\right)\\\frac{13}{203}\times 45-\frac{3}{203}\left(-8\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1\\3\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=1,y=3$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua o te whārite ki te 6, arā, te tauraro pātahi he tino iti rawa te kitea o 3,2.

$$2\left(9x+4y\right)-3\left(5x-11\right)=78-6y$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te 9x+4y.

$$18x+8y-3\left(5x-11\right)=78-6y$$

Whakamahia te āhuatanga tohatoha hei whakarea te -3 ki te 5x-11.

$$18x+8y-15x+33=78-6y$$

Pahekotia te 18x me -15x, ka 3x.

$$3x+8y+33=78-6y$$

Me tāpiri te 6y ki ngā taha e rua.

$$3x+8y+33+6y=78$$

Pahekotia te 8y me 6y, ka 14y.

$$3x+14y+33=78$$

Tangohia te 33 mai i ngā taha e rua.

$$3x+14y=78-33$$

Tangohia te 33 i te 78, ka 45.

$$3x+14y=45$$

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.

$$3x+14y=45,13x-7y=-8$$

Kia ōrite ai a 3x me 13x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 13 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$13\times 3x+13\times 14y=13\times 45,3\times 13x+3\left(-7\right)y=3\left(-8\right)$$

Whakarūnātia.

$$39x+182y=585,39x-21y=-24$$

Me tango 39x-21y=-24 mai i 39x+182y=585 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$39x-39x+182y+21y=585+24$$

Tāpiri 39x ki te -39x. Ka whakakore atu ngā kupu 39x me -39x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$182y+21y=585+24$$

Tāpiri 182y ki te 21y.

$$203y=585+24$$

Tāpiri 585 ki te 24.

$$203y=609$$

Whakawehea ngā taha e rua ki te 203.

$$y=3$$

Whakaurua te 3 mō y ki 13x-7y=-8. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$13x-7\times 3=-8$$

Whakareatia -7 ki te 3.

$$13x-21=-8$$

Me tāpiri 21 ki ngā taha e rua o te whārite.

$$13x=13$$

Whakawehea ngā taha e rua ki te 13.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=3$$