Whakaoti i te {l}{2x-y+3/3-x-2y+3/4=4}{3x-4y+3/4+4x-2y-9/3=4}

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

Whakaoti mō x, y

$$x=7$$

$$y=5$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua o te whārite ki te 12, arā, te tauraro pātahi he tino iti rawa te kitea o 3,4.

$$4\left(2x-y+3\right)-3\left(x-2y+3\right)=48$$

Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te 2x-y+3.

$$8x-4y+12-3\left(x-2y+3\right)=48$$

Whakamahia te āhuatanga tohatoha hei whakarea te -3 ki te x-2y+3.

$$8x-4y+12-3x+6y-9=48$$

Pahekotia te 8x me -3x, ka 5x.

$$5x-4y+12+6y-9=48$$

Pahekotia te -4y me 6y, ka 2y.

$$5x+2y+12-9=48$$

Tangohia te 9 i te 12, ka 3.

$$5x+2y+3=48$$

Tangohia te 3 mai i ngā taha e rua.

$$5x+2y=48-3$$

Tangohia te 3 i te 48, ka 45.

$$5x+2y=45$$

Whakaarohia te whārite tuarua. Me whakarea ngā taha e rua o te whārite ki te 12, arā, te tauraro pātahi he tino iti rawa te kitea o 4,3.

$$3\left(3x-4y+3\right)+4\left(4x-2y-9\right)=48$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te 3x-4y+3.

$$9x-12y+9+4\left(4x-2y-9\right)=48$$

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

$$9x-12y+9+16x-8y-36=48$$

Pahekotia te 9x me 16x, ka 25x.

$$25x-12y+9-8y-36=48$$

Pahekotia te -12y me -8y, ka -20y.

$$25x-20y+9-36=48$$

Tangohia te 36 i te 9, ka -27.

$$25x-20y-27=48$$

Me tāpiri te 27 ki ngā taha e rua.

$$25x-20y=48+27$$

Tāpirihia te 48 ki te 27, ka 75.

$$25x-20y=75$$

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.

$$5x+2y=45,25x-20y=75$$

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

$$5x+2y=45$$

Me tango 2y mai i ngā taha e rua o te whārite.

$$5x=-2y+45$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\left(-2y+45\right)$$

Whakareatia \frac{1}{5} ki te -2y+45.

$$x=-\frac{2}{5}y+9$$

Whakakapia te -\frac{2y}{5}+9 mō te x ki tērā atu whārite, 25x-20y=75.

$$25\left(-\frac{2}{5}y+9\right)-20y=75$$

Whakareatia 25 ki te -\frac{2y}{5}+9.

$$-10y+225-20y=75$$

Tāpiri -10y ki te -20y.

$$-30y+225=75$$

Me tango 225 mai i ngā taha e rua o te whārite.

$$-30y=-150$$

Whakawehea ngā taha e rua ki te -30.

$$y=5$$

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

$$x=-\frac{2}{5}\times 5+9$$

Whakareatia -\frac{2}{5} ki te 5.

$$x=-2+9$$

Tāpiri 9 ki te -2.

$$x=7$$

Kua oti te pūnaha te whakatau.

$$x=7,y=5$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua o te whārite ki te 12, arā, te tauraro pātahi he tino iti rawa te kitea o 3,4.

$$4\left(2x-y+3\right)-3\left(x-2y+3\right)=48$$

Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te 2x-y+3.

$$8x-4y+12-3\left(x-2y+3\right)=48$$

Whakamahia te āhuatanga tohatoha hei whakarea te -3 ki te x-2y+3.

$$8x-4y+12-3x+6y-9=48$$

Pahekotia te 8x me -3x, ka 5x.

$$5x-4y+12+6y-9=48$$

Pahekotia te -4y me 6y, ka 2y.

$$5x+2y+12-9=48$$

Tangohia te 9 i te 12, ka 3.

$$5x+2y+3=48$$

Tangohia te 3 mai i ngā taha e rua.

$$5x+2y=48-3$$

Tangohia te 3 i te 48, ka 45.

$$5x+2y=45$$

Whakaarohia te whārite tuarua. Me whakarea ngā taha e rua o te whārite ki te 12, arā, te tauraro pātahi he tino iti rawa te kitea o 4,3.

$$3\left(3x-4y+3\right)+4\left(4x-2y-9\right)=48$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te 3x-4y+3.

$$9x-12y+9+4\left(4x-2y-9\right)=48$$

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

$$9x-12y+9+16x-8y-36=48$$

Pahekotia te 9x me 16x, ka 25x.

$$25x-12y+9-8y-36=48$$

Pahekotia te -12y me -8y, ka -20y.

$$25x-20y+9-36=48$$

Tangohia te 36 i te 9, ka -27.

$$25x-20y-27=48$$

Me tāpiri te 27 ki ngā taha e rua.

$$25x-20y=48+27$$

Tāpirihia te 48 ki te 27, ka 75.

$$25x-20y=75$$

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

$$5x+2y=45,25x-20y=75$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&2\\25&-20\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}45\\75\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&2\\25&-20\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&2\\25&-20\end{matrix}\right))\left(\begin{matrix}5&2\\25&-20\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&2\\25&-20\end{matrix}\right))\left(\begin{matrix}45\\75\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&2\\25&-20\end{matrix}\right))\left(\begin{matrix}45\\75\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&2\\25&-20\end{matrix}\right))\left(\begin{matrix}45\\75\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{20}{5\left(-20\right)-2\times 25}&-\frac{2}{5\left(-20\right)-2\times 25}\\-\frac{25}{5\left(-20\right)-2\times 25}&\frac{5}{5\left(-20\right)-2\times 25}\end{matrix}\right)\left(\begin{matrix}45\\75\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{15}&\frac{1}{75}\\\frac{1}{6}&-\frac{1}{30}\end{matrix}\right)\left(\begin{matrix}45\\75\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{15}\times 45+\frac{1}{75}\times 75\\\frac{1}{6}\times 45-\frac{1}{30}\times 75\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}7\\5\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=7,y=5$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua o te whārite ki te 12, arā, te tauraro pātahi he tino iti rawa te kitea o 3,4.

$$4\left(2x-y+3\right)-3\left(x-2y+3\right)=48$$

Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te 2x-y+3.

$$8x-4y+12-3\left(x-2y+3\right)=48$$

Whakamahia te āhuatanga tohatoha hei whakarea te -3 ki te x-2y+3.

$$8x-4y+12-3x+6y-9=48$$

Pahekotia te 8x me -3x, ka 5x.

$$5x-4y+12+6y-9=48$$

Pahekotia te -4y me 6y, ka 2y.

$$5x+2y+12-9=48$$

Tangohia te 9 i te 12, ka 3.

$$5x+2y+3=48$$

Tangohia te 3 mai i ngā taha e rua.

$$5x+2y=48-3$$

Tangohia te 3 i te 48, ka 45.

$$5x+2y=45$$

Whakaarohia te whārite tuarua. Me whakarea ngā taha e rua o te whārite ki te 12, arā, te tauraro pātahi he tino iti rawa te kitea o 4,3.

$$3\left(3x-4y+3\right)+4\left(4x-2y-9\right)=48$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te 3x-4y+3.

$$9x-12y+9+4\left(4x-2y-9\right)=48$$

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

$$9x-12y+9+16x-8y-36=48$$

Pahekotia te 9x me 16x, ka 25x.

$$25x-12y+9-8y-36=48$$

Pahekotia te -12y me -8y, ka -20y.

$$25x-20y+9-36=48$$

Tangohia te 36 i te 9, ka -27.

$$25x-20y-27=48$$

Me tāpiri te 27 ki ngā taha e rua.

$$25x-20y=48+27$$

Tāpirihia te 48 ki te 27, ka 75.

$$25x-20y=75$$

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.

$$5x+2y=45,25x-20y=75$$

Kia ōrite ai a 5x me 25x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 25 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 5.

$$25\times 5x+25\times 2y=25\times 45,5\times 25x+5\left(-20\right)y=5\times 75$$

Whakarūnātia.

$$125x+50y=1125,125x-100y=375$$

Me tango 125x-100y=375 mai i 125x+50y=1125 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$125x-125x+50y+100y=1125-375$$

Tāpiri 125x ki te -125x. Ka whakakore atu ngā kupu 125x me -125x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$50y+100y=1125-375$$

Tāpiri 50y ki te 100y.

$$150y=1125-375$$

Tāpiri 1125 ki te -375.

$$150y=750$$

Whakawehea ngā taha e rua ki te 150.

$$y=5$$

Whakaurua te 5 mō y ki 25x-20y=75. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$25x-20\times 5=75$$

Whakareatia -20 ki te 5.

$$25x-100=75$$

Me tāpiri 100 ki ngā taha e rua o te whārite.

$$25x=175$$

Whakawehea ngā taha e rua ki te 25.

$$x=7$$

Kua oti te pūnaha te whakatau.

$$x=7,y=5$$