Whakaoti i te {l}{3x-7/4-2y+1/6=0}{x+2/5-5y+4/3=-2}

$$\left\{ \begin{array} { l } { \frac { 3 x - 7 } { 4 } - \frac { 2 y + 1 } { 6 } = 0 } \\ { \frac { x + 2 } { 5 } - \frac { 5 y + 4 } { 3 } = - 2 } \end{array} \right.$$

Whakaoti mō x, y

$$x=3$$

$$y=1$$

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 4,6.

$$3\left(3x-7\right)-2\left(2y+1\right)=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te 3x-7.

$$9x-21-2\left(2y+1\right)=0$$

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

$$9x-21-4y-2=0$$

Tangohia te 2 i te -21, ka -23.

$$9x-23-4y=0$$

Me tāpiri te 23 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$9x-4y=23$$

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.

$$3\left(x+2\right)-5\left(5y+4\right)=-30$$

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

$$3x+6-5\left(5y+4\right)=-30$$

Whakamahia te āhuatanga tohatoha hei whakarea te -5 ki te 5y+4.

$$3x+6-25y-20=-30$$

Tangohia te 20 i te 6, ka -14.

$$3x-14-25y=-30$$

Me tāpiri te 14 ki ngā taha e rua.

$$3x-25y=-30+14$$

Tāpirihia te -30 ki te 14, ka -16.

$$3x-25y=-16$$

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.

$$9x-4y=23,3x-25y=-16$$

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

$$9x-4y=23$$

Me tāpiri 4y ki ngā taha e rua o te whārite.

$$9x=4y+23$$

Whakawehea ngā taha e rua ki te 9.

$$x=\frac{1}{9}\left(4y+23\right)$$

Whakareatia \frac{1}{9} ki te 4y+23.

$$x=\frac{4}{9}y+\frac{23}{9}$$

Whakakapia te \frac{4y+23}{9} mō te x ki tērā atu whārite, 3x-25y=-16.

$$3\left(\frac{4}{9}y+\frac{23}{9}\right)-25y=-16$$

Whakareatia 3 ki te \frac{4y+23}{9}.

$$\frac{4}{3}y+\frac{23}{3}-25y=-16$$

Tāpiri \frac{4y}{3} ki te -25y.

$$-\frac{71}{3}y+\frac{23}{3}=-16$$

Me tango \frac{23}{3} mai i ngā taha e rua o te whārite.

$$-\frac{71}{3}y=-\frac{71}{3}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{71}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=1$$

Whakaurua te 1 mō y ki x=\frac{4}{9}y+\frac{23}{9}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{4+23}{9}$$

Tāpiri \frac{23}{9} ki te \frac{4}{9} 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.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=1$$

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 4,6.

$$3\left(3x-7\right)-2\left(2y+1\right)=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te 3x-7.

$$9x-21-2\left(2y+1\right)=0$$

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

$$9x-21-4y-2=0$$

Tangohia te 2 i te -21, ka -23.

$$9x-23-4y=0$$

Me tāpiri te 23 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$9x-4y=23$$

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.

$$3\left(x+2\right)-5\left(5y+4\right)=-30$$

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

$$3x+6-5\left(5y+4\right)=-30$$

Whakamahia te āhuatanga tohatoha hei whakarea te -5 ki te 5y+4.

$$3x+6-25y-20=-30$$

Tangohia te 20 i te 6, ka -14.

$$3x-14-25y=-30$$

Me tāpiri te 14 ki ngā taha e rua.

$$3x-25y=-30+14$$

Tāpirihia te -30 ki te 14, ka -16.

$$3x-25y=-16$$

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

$$9x-4y=23,3x-25y=-16$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}9&-4\\3&-25\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}23\\-16\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}9&-4\\3&-25\end{matrix}\right).

$$inverse(\left(\begin{matrix}9&-4\\3&-25\end{matrix}\right))\left(\begin{matrix}9&-4\\3&-25\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}9&-4\\3&-25\end{matrix}\right))\left(\begin{matrix}23\\-16\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}9&-4\\3&-25\end{matrix}\right))\left(\begin{matrix}23\\-16\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}9&-4\\3&-25\end{matrix}\right))\left(\begin{matrix}23\\-16\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te poukapa kōaro ko \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), kia tuhia anō ai te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{25}{9\left(-25\right)-\left(-4\times 3\right)}&-\frac{-4}{9\left(-25\right)-\left(-4\times 3\right)}\\-\frac{3}{9\left(-25\right)-\left(-4\times 3\right)}&\frac{9}{9\left(-25\right)-\left(-4\times 3\right)}\end{matrix}\right)\left(\begin{matrix}23\\-16\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{25}{213}&-\frac{4}{213}\\\frac{1}{71}&-\frac{3}{71}\end{matrix}\right)\left(\begin{matrix}23\\-16\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{25}{213}\times 23-\frac{4}{213}\left(-16\right)\\\frac{1}{71}\times 23-\frac{3}{71}\left(-16\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=3,y=1$$

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 4,6.

$$3\left(3x-7\right)-2\left(2y+1\right)=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te 3x-7.

$$9x-21-2\left(2y+1\right)=0$$

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

$$9x-21-4y-2=0$$

Tangohia te 2 i te -21, ka -23.

$$9x-23-4y=0$$

Me tāpiri te 23 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$9x-4y=23$$

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.

$$3\left(x+2\right)-5\left(5y+4\right)=-30$$

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

$$3x+6-5\left(5y+4\right)=-30$$

Whakamahia te āhuatanga tohatoha hei whakarea te -5 ki te 5y+4.

$$3x+6-25y-20=-30$$

Tangohia te 20 i te 6, ka -14.

$$3x-14-25y=-30$$

Me tāpiri te 14 ki ngā taha e rua.

$$3x-25y=-30+14$$

Tāpirihia te -30 ki te 14, ka -16.

$$3x-25y=-16$$

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.

$$9x-4y=23,3x-25y=-16$$

Kia ōrite ai a 9x 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 9.

$$3\times 9x+3\left(-4\right)y=3\times 23,9\times 3x+9\left(-25\right)y=9\left(-16\right)$$

Whakarūnātia.

$$27x-12y=69,27x-225y=-144$$

Me tango 27x-225y=-144 mai i 27x-12y=69 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$27x-27x-12y+225y=69+144$$

Tāpiri 27x ki te -27x. Ka whakakore atu ngā kupu 27x me -27x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-12y+225y=69+144$$

Tāpiri -12y ki te 225y.

$$213y=69+144$$

Tāpiri 69 ki te 144.

$$213y=213$$

Whakawehea ngā taha e rua ki te 213.

$$y=1$$

Whakaurua te 1 mō y ki 3x-25y=-16. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x-25=-16$$

Me tāpiri 25 ki ngā taha e rua o te whārite.

$$3x=9$$

Whakawehea ngā taha e rua ki te 3.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=1$$