Whakaoti i te {l}{4(x+y)-3(x-y)=10}{2(x+y)-3(x-y)=2}

$$\left\{ \begin{array} { l } { 4 ( x + y ) - 3 ( x - y ) = 10 } \\ { 2 ( x + y ) - 3 ( x - y ) = 2 } \end{array} \right.$$

Whakaoti mō x, y

$$x=3$$

$$y=1$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te x+y.

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

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

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

Pahekotia te 4x me -3x, ka x.

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

Pahekotia te 4y me 3y, ka 7y.

$$x+7y=10$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te x+y.

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

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

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

Pahekotia te 2x me -3x, ka -x.

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

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

$$-x+5y=2$$

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.

$$x+7y=10,-x+5y=2$$

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

$$x+7y=10$$

Me tango 7y mai i ngā taha e rua o te whārite.

$$x=-7y+10$$

Whakakapia te -7y+10 mō te x ki tērā atu whārite, -x+5y=2.

$$-\left(-7y+10\right)+5y=2$$

Whakareatia -1 ki te -7y+10.

$$7y-10+5y=2$$

Tāpiri 7y ki te 5y.

$$12y-10=2$$

Me tāpiri 10 ki ngā taha e rua o te whārite.

$$12y=12$$

Whakawehea ngā taha e rua ki te 12.

$$y=1$$

Whakaurua te 1 mō y ki x=-7y+10. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-7+10$$

Tāpiri 10 ki te -7.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=1$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te x+y.

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

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

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

Pahekotia te 4x me -3x, ka x.

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

Pahekotia te 4y me 3y, ka 7y.

$$x+7y=10$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te x+y.

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

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

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

Pahekotia te 2x me -3x, ka -x.

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

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

$$-x+5y=2$$

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

$$x+7y=10,-x+5y=2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&7\\-1&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}10\\2\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&7\\-1&5\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&7\\-1&5\end{matrix}\right))\left(\begin{matrix}1&7\\-1&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&7\\-1&5\end{matrix}\right))\left(\begin{matrix}10\\2\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&7\\-1&5\end{matrix}\right))\left(\begin{matrix}10\\2\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&7\\-1&5\end{matrix}\right))\left(\begin{matrix}10\\2\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{5-7\left(-1\right)}&-\frac{7}{5-7\left(-1\right)}\\-\frac{-1}{5-7\left(-1\right)}&\frac{1}{5-7\left(-1\right)}\end{matrix}\right)\left(\begin{matrix}10\\2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{12}&-\frac{7}{12}\\\frac{1}{12}&\frac{1}{12}\end{matrix}\right)\left(\begin{matrix}10\\2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{12}\times 10-\frac{7}{12}\times 2\\\frac{1}{12}\times 10+\frac{1}{12}\times 2\end{matrix}\right)$$

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

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

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

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

Pahekotia te 4x me -3x, ka x.

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

Pahekotia te 4y me 3y, ka 7y.

$$x+7y=10$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te x+y.

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

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

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

Pahekotia te 2x me -3x, ka -x.

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

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

$$-x+5y=2$$

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.

$$x+7y=10,-x+5y=2$$

Kia ōrite ai a x me -x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -1 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$-x-7y=-10,-x+5y=2$$

Me tango -x+5y=2 mai i -x-7y=-10 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-x+x-7y-5y=-10-2$$

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

$$-7y-5y=-10-2$$

Tāpiri -7y ki te -5y.

$$-12y=-10-2$$

Tāpiri -10 ki te -2.

$$-12y=-12$$

Whakawehea ngā taha e rua ki te -12.

$$y=1$$

Whakaurua te 1 mō y ki -x+5y=2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-x+5=2$$

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

$$-x=-3$$

Whakawehea ngā taha e rua ki te -1.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=1$$