Whakaoti i te {l}{x+1y-1/2+y-1/3=9}{x-1/3+y+1/2=8}

$$\left. \begin{array} { l } { \frac { x + 1 y - 1 } { 2 } + \frac { y - 1 } { 3 } = 9 } \\ { \frac { x - 1 } { 3 } + \frac { y + 1 } { 2 } = 8 } \end{array} \right.$$

Whakaoti mō x, y

$$x=58$$

$$y=-23$$

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 2,3.

$$3\left(x+1y-1\right)+2\left(y-1\right)=54$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te x+1y-1.

$$3x+3y-3+2\left(y-1\right)=54$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te y-1.

$$3x+3y-3+2y-2=54$$

Pahekotia te 3y me 2y, ka 5y.

$$3x+5y-3-2=54$$

Tangohia te 2 i te -3, ka -5.

$$3x+5y-5=54$$

Me tāpiri te 5 ki ngā taha e rua.

$$3x+5y=54+5$$

Tāpirihia te 54 ki te 5, ka 59.

$$3x+5y=59$$

Whakaarohia te whārite tuarua. 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.

$$2\left(x-1\right)+3\left(y+1\right)=48$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te x-1.

$$2x-2+3\left(y+1\right)=48$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te y+1.

$$2x-2+3y+3=48$$

Tāpirihia te -2 ki te 3, ka 1.

$$2x+1+3y=48$$

Tangohia te 1 mai i ngā taha e rua.

$$2x+3y=48-1$$

Tangohia te 1 i te 48, ka 47.

$$2x+3y=47$$

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.

$$3x+5y=59,2x+3y=47$$

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

$$3x+5y=59$$

Me tango 5y mai i ngā taha e rua o te whārite.

$$3x=-5y+59$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(-5y+59\right)$$

Whakareatia \frac{1}{3} ki te -5y+59.

$$x=-\frac{5}{3}y+\frac{59}{3}$$

Whakakapia te \frac{-5y+59}{3} mō te x ki tērā atu whārite, 2x+3y=47.

$$2\left(-\frac{5}{3}y+\frac{59}{3}\right)+3y=47$$

Whakareatia 2 ki te \frac{-5y+59}{3}.

$$-\frac{10}{3}y+\frac{118}{3}+3y=47$$

Tāpiri -\frac{10y}{3} ki te 3y.

$$-\frac{1}{3}y+\frac{118}{3}=47$$

Me tango \frac{118}{3} mai i ngā taha e rua o te whārite.

$$-\frac{1}{3}y=\frac{23}{3}$$

Me whakarea ngā taha e rua ki te -3.

$$y=-23$$

Whakaurua te -23 mō y ki x=-\frac{5}{3}y+\frac{59}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{5}{3}\left(-23\right)+\frac{59}{3}$$

Whakareatia -\frac{5}{3} ki te -23.

$$x=\frac{115+59}{3}$$

Tāpiri \frac{59}{3} ki te \frac{115}{3} 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.

$$x=58$$

Kua oti te pūnaha te whakatau.

$$x=58,y=-23$$

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 2,3.

$$3\left(x+1y-1\right)+2\left(y-1\right)=54$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te x+1y-1.

$$3x+3y-3+2\left(y-1\right)=54$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te y-1.

$$3x+3y-3+2y-2=54$$

Pahekotia te 3y me 2y, ka 5y.

$$3x+5y-3-2=54$$

Tangohia te 2 i te -3, ka -5.

$$3x+5y-5=54$$

Me tāpiri te 5 ki ngā taha e rua.

$$3x+5y=54+5$$

Tāpirihia te 54 ki te 5, ka 59.

$$3x+5y=59$$

Whakaarohia te whārite tuarua. 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.

$$2\left(x-1\right)+3\left(y+1\right)=48$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te x-1.

$$2x-2+3\left(y+1\right)=48$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te y+1.

$$2x-2+3y+3=48$$

Tāpirihia te -2 ki te 3, ka 1.

$$2x+1+3y=48$$

Tangohia te 1 mai i ngā taha e rua.

$$2x+3y=48-1$$

Tangohia te 1 i te 48, ka 47.

$$2x+3y=47$$

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

$$3x+5y=59,2x+3y=47$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&5\\2&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}59\\47\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}3&5\\2&3\end{matrix}\right))\left(\begin{matrix}3&5\\2&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&5\\2&3\end{matrix}\right))\left(\begin{matrix}59\\47\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&5\\2&3\end{matrix}\right))\left(\begin{matrix}59\\47\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&5\\2&3\end{matrix}\right))\left(\begin{matrix}59\\47\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{3\times 3-5\times 2}&-\frac{5}{3\times 3-5\times 2}\\-\frac{2}{3\times 3-5\times 2}&\frac{3}{3\times 3-5\times 2}\end{matrix}\right)\left(\begin{matrix}59\\47\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-3&5\\2&-3\end{matrix}\right)\left(\begin{matrix}59\\47\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-3\times 59+5\times 47\\2\times 59-3\times 47\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}58\\-23\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=58,y=-23$$

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 2,3.

$$3\left(x+1y-1\right)+2\left(y-1\right)=54$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te x+1y-1.

$$3x+3y-3+2\left(y-1\right)=54$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te y-1.

$$3x+3y-3+2y-2=54$$

Pahekotia te 3y me 2y, ka 5y.

$$3x+5y-3-2=54$$

Tangohia te 2 i te -3, ka -5.

$$3x+5y-5=54$$

Me tāpiri te 5 ki ngā taha e rua.

$$3x+5y=54+5$$

Tāpirihia te 54 ki te 5, ka 59.

$$3x+5y=59$$

Whakaarohia te whārite tuarua. 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.

$$2\left(x-1\right)+3\left(y+1\right)=48$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te x-1.

$$2x-2+3\left(y+1\right)=48$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te y+1.

$$2x-2+3y+3=48$$

Tāpirihia te -2 ki te 3, ka 1.

$$2x+1+3y=48$$

Tangohia te 1 mai i ngā taha e rua.

$$2x+3y=48-1$$

Tangohia te 1 i te 48, ka 47.

$$2x+3y=47$$

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.

$$3x+5y=59,2x+3y=47$$

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.

$$2\times 3x+2\times 5y=2\times 59,3\times 2x+3\times 3y=3\times 47$$

Whakarūnātia.

$$6x+10y=118,6x+9y=141$$

Me tango 6x+9y=141 mai i 6x+10y=118 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x+10y-9y=118-141$$

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.

$$10y-9y=118-141$$

Tāpiri 10y ki te -9y.

$$y=118-141$$

Tāpiri 118 ki te -141.

$$y=-23$$

Whakaurua te -23 mō y ki 2x+3y=47. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x+3\left(-23\right)=47$$

Whakareatia 3 ki te -23.

$$2x-69=47$$

Me tāpiri 69 ki ngā taha e rua o te whārite.

$$2x=116$$

Whakawehea ngā taha e rua ki te 2.

$$x=58$$

Kua oti te pūnaha te whakatau.

$$x=58,y=-23$$