Whakaoti i te {l}{0,2x-0,3(2y+1)=1,5}{3(x+1)+3y=2y-2}

$$\left\{ \begin{array} { l } { 0,2 x - 0,3 ( 2 y + 1 ) = 1,5 } \\ { 3 ( x + 1 ) + 3 y = 2 y - 2 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-0.6$$

$$y=-3.2$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te -0.3 ki te 2y+1.

$$0.2x-0.6y-0.3=1.5$$

Me tāpiri te 0.3 ki ngā taha e rua.

$$0.2x-0.6y=1.5+0.3$$

Tāpirihia te 1.5 ki te 0.3, ka 1.8.

$$0.2x-0.6y=1.8$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te x+1.

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

Tangohia te 2y mai i ngā taha e rua.

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

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

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

Tangohia te 3 mai i ngā taha e rua.

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

Tangohia te 3 i te -2, ka -5.

$$3x+y=-5$$

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.

$$0.2x-0.6y=1.8,3x+y=-5$$

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

$$0.2x-0.6y=1.8$$

Me tāpiri \frac{3y}{5} ki ngā taha e rua o te whārite.

$$0.2x=0.6y+1.8$$

Me whakarea ngā taha e rua ki te 5.

$$x=5\left(0.6y+1.8\right)$$

Whakareatia 5 ki te \frac{3y+9}{5}.

$$x=3y+9$$

Whakakapia te 9+3y mō te x ki tērā atu whārite, 3x+y=-5.

$$3\left(3y+9\right)+y=-5$$

Whakareatia 3 ki te 9+3y.

$$9y+27+y=-5$$

Tāpiri 9y ki te y.

$$10y+27=-5$$

Me tango 27 mai i ngā taha e rua o te whārite.

$$10y=-32$$

Whakawehea ngā taha e rua ki te 10.

$$y=-\frac{16}{5}$$

Whakaurua te -\frac{16}{5} mō y ki x=3y+9. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=3\left(-\frac{16}{5}\right)+9$$

Whakareatia 3 ki te -\frac{16}{5}.

$$x=-\frac{48}{5}+9$$

Tāpiri 9 ki te -\frac{48}{5}.

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

Kua oti te pūnaha te whakatau.

$$x=-\frac{3}{5},y=-\frac{16}{5}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te -0.3 ki te 2y+1.

$$0.2x-0.6y-0.3=1.5$$

Me tāpiri te 0.3 ki ngā taha e rua.

$$0.2x-0.6y=1.5+0.3$$

Tāpirihia te 1.5 ki te 0.3, ka 1.8.

$$0.2x-0.6y=1.8$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te x+1.

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

Tangohia te 2y mai i ngā taha e rua.

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

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

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

Tangohia te 3 mai i ngā taha e rua.

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

Tangohia te 3 i te -2, ka -5.

$$3x+y=-5$$

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

$$0.2x-0.6y=1.8,3x+y=-5$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}0.2&-0.6\\3&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1.8\\-5\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}0.2&-0.6\\3&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}0.2&-0.6\\3&1\end{matrix}\right))\left(\begin{matrix}0.2&-0.6\\3&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}0.2&-0.6\\3&1\end{matrix}\right))\left(\begin{matrix}1.8\\-5\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}0.2&-0.6\\3&1\end{matrix}\right))\left(\begin{matrix}1.8\\-5\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}0.2&-0.6\\3&1\end{matrix}\right))\left(\begin{matrix}1.8\\-5\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{0.2-\left(-0.6\times 3\right)}&-\frac{-0.6}{0.2-\left(-0.6\times 3\right)}\\-\frac{3}{0.2-\left(-0.6\times 3\right)}&\frac{0.2}{0.2-\left(-0.6\times 3\right)}\end{matrix}\right)\left(\begin{matrix}1.8\\-5\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}&\frac{3}{10}\\-\frac{3}{2}&\frac{1}{10}\end{matrix}\right)\left(\begin{matrix}1.8\\-5\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}\times 1.8+\frac{3}{10}\left(-5\right)\\-\frac{3}{2}\times 1.8+\frac{1}{10}\left(-5\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{5}\\-\frac{16}{5}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{3}{5},y=-\frac{16}{5}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te -0.3 ki te 2y+1.

$$0.2x-0.6y-0.3=1.5$$

Me tāpiri te 0.3 ki ngā taha e rua.

$$0.2x-0.6y=1.5+0.3$$

Tāpirihia te 1.5 ki te 0.3, ka 1.8.

$$0.2x-0.6y=1.8$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te x+1.

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

Tangohia te 2y mai i ngā taha e rua.

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

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

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

Tangohia te 3 mai i ngā taha e rua.

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

Tangohia te 3 i te -2, ka -5.

$$3x+y=-5$$

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.

$$0.2x-0.6y=1.8,3x+y=-5$$

Kia ōrite ai a \frac{x}{5} 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 0.2.

$$3\times 0.2x+3\left(-0.6\right)y=3\times 1.8,0.2\times 3x+0.2y=0.2\left(-5\right)$$

Whakarūnātia.

$$0.6x-1.8y=5.4,0.6x+0.2y=-1$$

Me tango 0.6x+0.2y=-1 mai i 0.6x-1.8y=5.4 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$0.6x-0.6x-1.8y-0.2y=5.4+1$$

Tāpiri \frac{3x}{5} ki te -\frac{3x}{5}. Ka whakakore atu ngā kupu \frac{3x}{5} me -\frac{3x}{5}, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-1.8y-0.2y=5.4+1$$

Tāpiri -\frac{9y}{5} ki te -\frac{y}{5}.

$$-2y=5.4+1$$

Tāpiri 5.4 ki te 1.

$$-2y=6.4$$

Whakawehea ngā taha e rua ki te -2.

$$y=-\frac{16}{5}$$

Whakaurua te -\frac{16}{5} mō y ki 3x+y=-5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x-\frac{16}{5}=-5$$

Me tāpiri \frac{16}{5} ki ngā taha e rua o te whārite.

$$3x=-\frac{9}{5}$$

Whakawehea ngā taha e rua ki te 3.

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

Kua oti te pūnaha te whakatau.

$$x=-\frac{3}{5},y=-\frac{16}{5}$$