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

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

Whakaoti mō x, y

$$x = -\frac{5}{2} = -2\frac{1}{2} = -2.5$$

$$y = -\frac{13}{10} = -1\frac{3}{10} = -1.3$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te x+y.

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

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

$$3x+3y+9=2x-2y$$

Tangohia te 2x mai i ngā taha e rua.

$$3x+3y+9-2x=-2y$$

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

$$x+3y+9=-2y$$

Me tāpiri te 2y ki ngā taha e rua.

$$x+3y+9+2y=0$$

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

$$x+5y+9=0$$

Tangohia te 9 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$x+5y=-9$$

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

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

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

$$2x+2y=3x-3y-4$$

Tangohia te 3x mai i ngā taha e rua.

$$2x+2y-3x=-3y-4$$

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

$$-x+2y=-3y-4$$

Me tāpiri te 3y ki ngā taha e rua.

$$-x+2y+3y=-4$$

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

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

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.

$$x+5y=-9,-x+5y=-4$$

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

$$x+5y=-9$$

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

$$x=-5y-9$$

Whakakapia te -5y-9 mō te x ki tērā atu whārite, -x+5y=-4.

$$-\left(-5y-9\right)+5y=-4$$

Whakareatia -1 ki te -5y-9.

$$5y+9+5y=-4$$

Tāpiri 5y ki te 5y.

$$10y+9=-4$$

Me tango 9 mai i ngā taha e rua o te whārite.

$$10y=-13$$

Whakawehea ngā taha e rua ki te 10.

$$y=-\frac{13}{10}$$

Whakaurua te -\frac{13}{10} mō y ki x=-5y-9. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-5\left(-\frac{13}{10}\right)-9$$

Whakareatia -5 ki te -\frac{13}{10}.

$$x=\frac{13}{2}-9$$

Tāpiri -9 ki te \frac{13}{2}.

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

Kua oti te pūnaha te whakatau.

$$x=-\frac{5}{2},y=-\frac{13}{10}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te x+y.

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

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

$$3x+3y+9=2x-2y$$

Tangohia te 2x mai i ngā taha e rua.

$$3x+3y+9-2x=-2y$$

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

$$x+3y+9=-2y$$

Me tāpiri te 2y ki ngā taha e rua.

$$x+3y+9+2y=0$$

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

$$x+5y+9=0$$

Tangohia te 9 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$x+5y=-9$$

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

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

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

$$2x+2y=3x-3y-4$$

Tangohia te 3x mai i ngā taha e rua.

$$2x+2y-3x=-3y-4$$

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

$$-x+2y=-3y-4$$

Me tāpiri te 3y ki ngā taha e rua.

$$-x+2y+3y=-4$$

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

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

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

$$x+5y=-9,-x+5y=-4$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&5\\-1&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-9\\-4\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&5\\-1&5\end{matrix}\right))\left(\begin{matrix}1&5\\-1&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&5\\-1&5\end{matrix}\right))\left(\begin{matrix}-9\\-4\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&5\\-1&5\end{matrix}\right))\left(\begin{matrix}-9\\-4\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&5\\-1&5\end{matrix}\right))\left(\begin{matrix}-9\\-4\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{5-5\left(-1\right)}&-\frac{5}{5-5\left(-1\right)}\\-\frac{-1}{5-5\left(-1\right)}&\frac{1}{5-5\left(-1\right)}\end{matrix}\right)\left(\begin{matrix}-9\\-4\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}&-\frac{1}{2}\\\frac{1}{10}&\frac{1}{10}\end{matrix}\right)\left(\begin{matrix}-9\\-4\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}\left(-9\right)-\frac{1}{2}\left(-4\right)\\\frac{1}{10}\left(-9\right)+\frac{1}{10}\left(-4\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{2}\\-\frac{13}{10}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{5}{2},y=-\frac{13}{10}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te x+y.

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

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

$$3x+3y+9=2x-2y$$

Tangohia te 2x mai i ngā taha e rua.

$$3x+3y+9-2x=-2y$$

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

$$x+3y+9=-2y$$

Me tāpiri te 2y ki ngā taha e rua.

$$x+3y+9+2y=0$$

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

$$x+5y+9=0$$

Tangohia te 9 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$x+5y=-9$$

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

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

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

$$2x+2y=3x-3y-4$$

Tangohia te 3x mai i ngā taha e rua.

$$2x+2y-3x=-3y-4$$

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

$$-x+2y=-3y-4$$

Me tāpiri te 3y ki ngā taha e rua.

$$-x+2y+3y=-4$$

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

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

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.

$$x+5y=-9,-x+5y=-4$$

Me tango -x+5y=-4 mai i x+5y=-9 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$x+x+5y-5y=-9+4$$

Tāpiri 5y ki te -5y. Ka whakakore atu ngā kupu 5y me -5y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$x+x=-9+4$$

Tāpiri x ki te x.

$$2x=-9+4$$

Tāpiri -9 ki te 4.

$$2x=-5$$

Whakawehea ngā taha e rua ki te 2.

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

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

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

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

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

Me tango \frac{5}{2} mai i ngā taha e rua o te whārite.

$$5y=-\frac{13}{2}$$

Whakawehea ngā taha e rua ki te 5.

$$y=-\frac{13}{10}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{5}{2},y=-\frac{13}{10}$$