Whakaoti i te {l}{2-y=2(6x+3)+y}{x+4=3y}

$$\left\{ \begin{array} { l } { 2 - y = 2 ( 6 x + 3 ) + y } \\ { x + 4 = 3 y } \end{array} \right.$$

Whakaoti mō y, x

$$x=-\frac{10}{19}\approx -0.526315789$$

$$y = \frac{22}{19} = 1\frac{3}{19} \approx 1.157894737$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te 6x+3.

$$2-y=12x+6+y$$

Tangohia te 12x mai i ngā taha e rua.

$$2-y-12x=6+y$$

Tangohia te y mai i ngā taha e rua.

$$2-y-12x-y=6$$

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

$$2-2y-12x=6$$

Tangohia te 2 mai i ngā taha e rua.

$$-2y-12x=6-2$$

Tangohia te 2 i te 6, ka 4.

$$-2y-12x=4$$

Whakaarohia te whārite tuarua. Tangohia te 3y mai i ngā taha e rua.

$$x+4-3y=0$$

Tangohia te 4 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$x-3y=-4$$

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.

$$-2y-12x=4,-3y+x=-4$$

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

$$-2y-12x=4$$

Me tāpiri 12x ki ngā taha e rua o te whārite.

$$-2y=12x+4$$

Whakawehea ngā taha e rua ki te -2.

$$y=-\frac{1}{2}\left(12x+4\right)$$

Whakareatia -\frac{1}{2} ki te 12x+4.

$$y=-6x-2$$

Whakakapia te -6x-2 mō te y ki tērā atu whārite, -3y+x=-4.

$$-3\left(-6x-2\right)+x=-4$$

Whakareatia -3 ki te -6x-2.

$$18x+6+x=-4$$

Tāpiri 18x ki te x.

$$19x+6=-4$$

Me tango 6 mai i ngā taha e rua o te whārite.

$$19x=-10$$

Whakawehea ngā taha e rua ki te 19.

$$x=-\frac{10}{19}$$

Whakaurua te -\frac{10}{19} mō x ki y=-6x-2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=-6\left(-\frac{10}{19}\right)-2$$

Whakareatia -6 ki te -\frac{10}{19}.

$$y=\frac{60}{19}-2$$

Tāpiri -2 ki te \frac{60}{19}.

$$y=\frac{22}{19}$$

Kua oti te pūnaha te whakatau.

$$y=\frac{22}{19},x=-\frac{10}{19}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te 6x+3.

$$2-y=12x+6+y$$

Tangohia te 12x mai i ngā taha e rua.

$$2-y-12x=6+y$$

Tangohia te y mai i ngā taha e rua.

$$2-y-12x-y=6$$

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

$$2-2y-12x=6$$

Tangohia te 2 mai i ngā taha e rua.

$$-2y-12x=6-2$$

Tangohia te 2 i te 6, ka 4.

$$-2y-12x=4$$

Whakaarohia te whārite tuarua. Tangohia te 3y mai i ngā taha e rua.

$$x+4-3y=0$$

Tangohia te 4 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$x-3y=-4$$

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

$$-2y-12x=4,-3y+x=-4$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-2&-12\\-3&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}4\\-4\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-2&-12\\-3&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}-2&-12\\-3&1\end{matrix}\right))\left(\begin{matrix}-2&-12\\-3&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}-2&-12\\-3&1\end{matrix}\right))\left(\begin{matrix}4\\-4\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}-2&-12\\-3&1\end{matrix}\right))\left(\begin{matrix}4\\-4\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}-2&-12\\-3&1\end{matrix}\right))\left(\begin{matrix}4\\-4\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{-2-\left(-12\left(-3\right)\right)}&-\frac{-12}{-2-\left(-12\left(-3\right)\right)}\\-\frac{-3}{-2-\left(-12\left(-3\right)\right)}&-\frac{2}{-2-\left(-12\left(-3\right)\right)}\end{matrix}\right)\left(\begin{matrix}4\\-4\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{38}&-\frac{6}{19}\\-\frac{3}{38}&\frac{1}{19}\end{matrix}\right)\left(\begin{matrix}4\\-4\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{38}\times 4-\frac{6}{19}\left(-4\right)\\-\frac{3}{38}\times 4+\frac{1}{19}\left(-4\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{22}{19}\\-\frac{10}{19}\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=\frac{22}{19},x=-\frac{10}{19}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te 6x+3.

$$2-y=12x+6+y$$

Tangohia te 12x mai i ngā taha e rua.

$$2-y-12x=6+y$$

Tangohia te y mai i ngā taha e rua.

$$2-y-12x-y=6$$

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

$$2-2y-12x=6$$

Tangohia te 2 mai i ngā taha e rua.

$$-2y-12x=6-2$$

Tangohia te 2 i te 6, ka 4.

$$-2y-12x=4$$

Whakaarohia te whārite tuarua. Tangohia te 3y mai i ngā taha e rua.

$$x+4-3y=0$$

Tangohia te 4 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$x-3y=-4$$

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.

$$-2y-12x=4,-3y+x=-4$$

Kia ōrite ai a -2y me -3y, 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 -2.

$$-3\left(-2\right)y-3\left(-12\right)x=-3\times 4,-2\left(-3\right)y-2x=-2\left(-4\right)$$

Whakarūnātia.

$$6y+36x=-12,6y-2x=8$$

Me tango 6y-2x=8 mai i 6y+36x=-12 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6y-6y+36x+2x=-12-8$$

Tāpiri 6y ki te -6y. Ka whakakore atu ngā kupu 6y me -6y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$36x+2x=-12-8$$

Tāpiri 36x ki te 2x.

$$38x=-12-8$$

Tāpiri -12 ki te -8.

$$38x=-20$$

Whakawehea ngā taha e rua ki te 38.

$$x=-\frac{10}{19}$$

Whakaurua te -\frac{10}{19} mō x ki -3y+x=-4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$-3y-\frac{10}{19}=-4$$

Me tāpiri \frac{10}{19} ki ngā taha e rua o te whārite.

$$-3y=-\frac{66}{19}$$

Whakawehea ngā taha e rua ki te -3.

$$y=\frac{22}{19}$$

Kua oti te pūnaha te whakatau.

$$y=\frac{22}{19},x=-\frac{10}{19}$$