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

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

Whakaoti mō x, y

$$x=\frac{16D+11}{4D+1}$$

$$y=-\frac{7}{4D+1}$$

$$D\neq -\frac{1}{4}$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

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

Tangohia te 3x mai i ngā taha e rua.

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

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

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

Tangohia te 4yD mai i ngā taha e rua.

$$-x-2y-3-4yD=0$$

Me tāpiri te 3 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

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

Pahekotia ngā kīanga tau katoa e whai ana i te x,y.

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

Whakaarohia te whārite tuarua. Whakareatia te 2 ki te 2, ka 4.

$$x+y=4$$

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.

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

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

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

Me tango -2y-4yD mai i ngā taha e rua o te whārite.

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

Whakawehea ngā taha e rua ki te -1.

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

Whakareatia -1 ki te 2y+4yD+3.

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

Whakakapia te -2y-4yD-3 mō te x ki tērā atu whārite, x+y=4.

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

Tāpiri -2y-4yD ki te y.

$$\left(-4D-1\right)y-3=4$$

Me tāpiri 3 ki ngā taha e rua o te whārite.

$$\left(-4D-1\right)y=7$$

Whakawehea ngā taha e rua ki te -1-4D.

$$y=-\frac{7}{4D+1}$$

Whakaurua te -\frac{7}{1+4D} mō y ki x=\left(-4D-2\right)y-3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\left(-4D-2\right)\left(-\frac{7}{4D+1}\right)-3$$

Whakareatia -2-4D ki te -\frac{7}{1+4D}.

$$x=\frac{14\left(2D+1\right)}{4D+1}-3$$

Tāpiri -3 ki te \frac{14\left(1+2D\right)}{1+4D}.

$$x=\frac{16D+11}{4D+1}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{16D+11}{4D+1},y=-\frac{7}{4D+1}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

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

Tangohia te 3x mai i ngā taha e rua.

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

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

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

Tangohia te 4yD mai i ngā taha e rua.

$$-x-2y-3-4yD=0$$

Me tāpiri te 3 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

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

Pahekotia ngā kīanga tau katoa e whai ana i te x,y.

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

Whakaarohia te whārite tuarua. Whakareatia te 2 ki te 2, ka 4.

$$x+y=4$$

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

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-1&-4D-2\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3\\4\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-1&-2-4D\\1&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}-1&-4D-2\\1&1\end{matrix}\right))\left(\begin{matrix}-1&-4D-2\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&-4D-2\\1&1\end{matrix}\right))\left(\begin{matrix}3\\4\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&-4D-2\\1&1\end{matrix}\right))\left(\begin{matrix}3\\4\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&-4D-2\\1&1\end{matrix}\right))\left(\begin{matrix}3\\4\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{-1-\left(-4D-2\right)}&-\frac{-4D-2}{-1-\left(-4D-2\right)}\\-\frac{1}{-1-\left(-4D-2\right)}&-\frac{1}{-1-\left(-4D-2\right)}\end{matrix}\right)\left(\begin{matrix}3\\4\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4D+1}&\frac{2\left(2D+1\right)}{4D+1}\\-\frac{1}{4D+1}&-\frac{1}{4D+1}\end{matrix}\right)\left(\begin{matrix}3\\4\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4D+1}\times 3+\frac{2\left(2D+1\right)}{4D+1}\times 4\\\left(-\frac{1}{4D+1}\right)\times 3+\left(-\frac{1}{4D+1}\right)\times 4\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{16D+11}{4D+1}\\-\frac{7}{4D+1}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{16D+11}{4D+1},y=-\frac{7}{4D+1}$$

Ngā Upane e Whakamahi i te Tangohanga

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

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

Tangohia te 3x mai i ngā taha e rua.

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

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

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

Tangohia te 4yD mai i ngā taha e rua.

$$-x-2y-3-4yD=0$$

Me tāpiri te 3 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

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

Pahekotia ngā kīanga tau katoa e whai ana i te x,y.

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

Whakaarohia te whārite tuarua. Whakareatia te 2 ki te 2, ka 4.

$$x+y=4$$

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.

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

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.

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

Me tango -x-y=-4 mai i -x+\left(-4D-2\right)y=3 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

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

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.

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

Tāpiri -2y-4yD ki te y.

$$\left(-4D-1\right)y=3+4$$

Tāpiri 3 ki te 4.

$$\left(-4D-1\right)y=7$$

Whakawehea ngā taha e rua ki te -1-4D.

$$y=-\frac{7}{4D+1}$$

Whakaurua te -\frac{7}{1+4D} mō y ki x+y=4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x-\frac{7}{4D+1}=4$$

Me tāpiri \frac{7}{1+4D} ki ngā taha e rua o te whārite.

$$x=\frac{16D+11}{4D+1}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{16D+11}{4D+1},y=-\frac{7}{4D+1}$$