Whakaoti i te {l}{x/3+y/4=5/6}{3x+20y/5-frac{8y+1}{3}=frac{12x+16y}{15}}

$$\left\{ \begin{array}{l}{ \frac { x } { 3 } + \frac { y } { 4 } = \frac { 5 } { 6 } }\\{ \frac { 3 x + 20 y } { 5 } - \frac { 8 y + 1 } { 3 } = \frac { 12 x + 16 y } { 15 } }\end{array} \right.$$

Whakaoti mō x, y

$$x=1$$

$$y=2$$

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 3,4,6.

$$4x+3y=10$$

Whakaarohia te whārite tuarua. Me whakarea ngā taha e rua o te whārite ki te 15, arā, te tauraro pātahi he tino iti rawa te kitea o 5,3,15.

$$3\left(3x+20y\right)-5\left(8y+1\right)=12x+16y$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te 3x+20y.

$$9x+60y-5\left(8y+1\right)=12x+16y$$

Whakamahia te āhuatanga tohatoha hei whakarea te -5 ki te 8y+1.

$$9x+60y-40y-5=12x+16y$$

Pahekotia te 60y me -40y, ka 20y.

$$9x+20y-5=12x+16y$$

Tangohia te 12x mai i ngā taha e rua.

$$9x+20y-5-12x=16y$$

Pahekotia te 9x me -12x, ka -3x.

$$-3x+20y-5=16y$$

Tangohia te 16y mai i ngā taha e rua.

$$-3x+20y-5-16y=0$$

Pahekotia te 20y me -16y, ka 4y.

$$-3x+4y-5=0$$

Me tāpiri te 5 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$-3x+4y=5$$

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.

$$4x+3y=10,-3x+4y=5$$

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

$$4x+3y=10$$

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

$$4x=-3y+10$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{1}{4}\left(-3y+10\right)$$

Whakareatia \frac{1}{4} ki te -3y+10.

$$x=-\frac{3}{4}y+\frac{5}{2}$$

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

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

Whakareatia -3 ki te -\frac{3y}{4}+\frac{5}{2}.

$$\frac{9}{4}y-\frac{15}{2}+4y=5$$

Tāpiri \frac{9y}{4} ki te 4y.

$$\frac{25}{4}y-\frac{15}{2}=5$$

Me tāpiri \frac{15}{2} ki ngā taha e rua o te whārite.

$$\frac{25}{4}y=\frac{25}{2}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{25}{4}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=2$$

Whakaurua te 2 mō y ki x=-\frac{3}{4}y+\frac{5}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{3}{4}\times 2+\frac{5}{2}$$

Whakareatia -\frac{3}{4} ki te 2.

$$x=\frac{-3+5}{2}$$

Tāpiri \frac{5}{2} ki te -\frac{3}{2} 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.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=2$$

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 3,4,6.

$$4x+3y=10$$

Whakaarohia te whārite tuarua. Me whakarea ngā taha e rua o te whārite ki te 15, arā, te tauraro pātahi he tino iti rawa te kitea o 5,3,15.

$$3\left(3x+20y\right)-5\left(8y+1\right)=12x+16y$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te 3x+20y.

$$9x+60y-5\left(8y+1\right)=12x+16y$$

Whakamahia te āhuatanga tohatoha hei whakarea te -5 ki te 8y+1.

$$9x+60y-40y-5=12x+16y$$

Pahekotia te 60y me -40y, ka 20y.

$$9x+20y-5=12x+16y$$

Tangohia te 12x mai i ngā taha e rua.

$$9x+20y-5-12x=16y$$

Pahekotia te 9x me -12x, ka -3x.

$$-3x+20y-5=16y$$

Tangohia te 16y mai i ngā taha e rua.

$$-3x+20y-5-16y=0$$

Pahekotia te 20y me -16y, ka 4y.

$$-3x+4y-5=0$$

Me tāpiri te 5 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$-3x+4y=5$$

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

$$4x+3y=10,-3x+4y=5$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&3\\-3&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}10\\5\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}4&3\\-3&4\end{matrix}\right))\left(\begin{matrix}4&3\\-3&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&3\\-3&4\end{matrix}\right))\left(\begin{matrix}10\\5\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&3\\-3&4\end{matrix}\right))\left(\begin{matrix}10\\5\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&3\\-3&4\end{matrix}\right))\left(\begin{matrix}10\\5\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{4\times 4-3\left(-3\right)}&-\frac{3}{4\times 4-3\left(-3\right)}\\-\frac{-3}{4\times 4-3\left(-3\right)}&\frac{4}{4\times 4-3\left(-3\right)}\end{matrix}\right)\left(\begin{matrix}10\\5\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{25}&-\frac{3}{25}\\\frac{3}{25}&\frac{4}{25}\end{matrix}\right)\left(\begin{matrix}10\\5\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{25}\times 10-\frac{3}{25}\times 5\\\frac{3}{25}\times 10+\frac{4}{25}\times 5\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1\\2\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=1,y=2$$

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 3,4,6.

$$4x+3y=10$$

Whakaarohia te whārite tuarua. Me whakarea ngā taha e rua o te whārite ki te 15, arā, te tauraro pātahi he tino iti rawa te kitea o 5,3,15.

$$3\left(3x+20y\right)-5\left(8y+1\right)=12x+16y$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te 3x+20y.

$$9x+60y-5\left(8y+1\right)=12x+16y$$

Whakamahia te āhuatanga tohatoha hei whakarea te -5 ki te 8y+1.

$$9x+60y-40y-5=12x+16y$$

Pahekotia te 60y me -40y, ka 20y.

$$9x+20y-5=12x+16y$$

Tangohia te 12x mai i ngā taha e rua.

$$9x+20y-5-12x=16y$$

Pahekotia te 9x me -12x, ka -3x.

$$-3x+20y-5=16y$$

Tangohia te 16y mai i ngā taha e rua.

$$-3x+20y-5-16y=0$$

Pahekotia te 20y me -16y, ka 4y.

$$-3x+4y-5=0$$

Me tāpiri te 5 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$-3x+4y=5$$

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.

$$4x+3y=10,-3x+4y=5$$

Kia ōrite ai a 4x 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 4.

$$-3\times 4x-3\times 3y=-3\times 10,4\left(-3\right)x+4\times 4y=4\times 5$$

Whakarūnātia.

$$-12x-9y=-30,-12x+16y=20$$

Me tango -12x+16y=20 mai i -12x-9y=-30 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-12x+12x-9y-16y=-30-20$$

Tāpiri -12x ki te 12x. Ka whakakore atu ngā kupu -12x me 12x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-9y-16y=-30-20$$

Tāpiri -9y ki te -16y.

$$-25y=-30-20$$

Tāpiri -30 ki te -20.

$$-25y=-50$$

Whakawehea ngā taha e rua ki te -25.

$$y=2$$

Whakaurua te 2 mō y ki -3x+4y=5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-3x+4\times 2=5$$

Whakareatia 4 ki te 2.

$$-3x+8=5$$

Me tango 8 mai i ngā taha e rua o te whārite.

$$-3x=-3$$

Whakawehea ngā taha e rua ki te -3.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=2$$