Whakaoti i te {l}{3(x+2)=2y}{2cy+5=7x}

$$\left\{ \begin{array} { l } { 3 ( x + 2 ) = 2 y } \\ { 2 c y + 5 = 7 x } \end{array} \right.$$

Whakaoti mō x, y

$$x=-\frac{6c+5}{3c-7}$$

$$y=-\frac{57}{2\left(3c-7\right)}$$

$$c\neq \frac{7}{3}$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

$$3x+6=2y$$

Tangohia te 2y mai i ngā taha e rua.

$$3x+6-2y=0$$

Tangohia te 6 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$3x-2y=-6$$

Whakaarohia te whārite tuarua. Tangohia te 7x mai i ngā taha e rua.

$$2cy+5-7x=0$$

Tangohia te 5 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$2cy-7x=-5$$

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.

$$3x-2y=-6,-7x+2cy=-5$$

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

$$3x-2y=-6$$

Me tāpiri 2y ki ngā taha e rua o te whārite.

$$3x=2y-6$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(2y-6\right)$$

Whakareatia \frac{1}{3} ki te -6+2y.

$$x=\frac{2}{3}y-2$$

Whakakapia te \frac{2y}{3}-2 mō te x ki tērā atu whārite, -7x+2cy=-5.

$$-7\left(\frac{2}{3}y-2\right)+2cy=-5$$

Whakareatia -7 ki te \frac{2y}{3}-2.

$$-\frac{14}{3}y+14+2cy=-5$$

Tāpiri -\frac{14y}{3} ki te 2cy.

$$\left(2c-\frac{14}{3}\right)y+14=-5$$

Me tango 14 mai i ngā taha e rua o te whārite.

$$\left(2c-\frac{14}{3}\right)y=-19$$

Whakawehea ngā taha e rua ki te -\frac{14}{3}+2c.

$$y=-\frac{57}{2\left(3c-7\right)}$$

Whakaurua te -\frac{57}{2\left(-7+3c\right)} mō y ki x=\frac{2}{3}y-2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{2}{3}\left(-\frac{57}{2\left(3c-7\right)}\right)-2$$

Whakareatia \frac{2}{3} ki te -\frac{57}{2\left(-7+3c\right)}.

$$x=-\frac{19}{3c-7}-2$$

Tāpiri -2 ki te -\frac{19}{-7+3c}.

$$x=-\frac{6c+5}{3c-7}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{6c+5}{3c-7},y=-\frac{57}{2\left(3c-7\right)}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$3x+6=2y$$

Tangohia te 2y mai i ngā taha e rua.

$$3x+6-2y=0$$

Tangohia te 6 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$3x-2y=-6$$

Whakaarohia te whārite tuarua. Tangohia te 7x mai i ngā taha e rua.

$$2cy+5-7x=0$$

Tangohia te 5 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$2cy-7x=-5$$

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

$$3x-2y=-6,-7x+2cy=-5$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&-2\\-7&2c\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-6\\-5\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&-2\\-7&2c\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&-2\\-7&2c\end{matrix}\right))\left(\begin{matrix}3&-2\\-7&2c\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-2\\-7&2c\end{matrix}\right))\left(\begin{matrix}-6\\-5\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-2\\-7&2c\end{matrix}\right))\left(\begin{matrix}-6\\-5\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-2\\-7&2c\end{matrix}\right))\left(\begin{matrix}-6\\-5\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2c}{3\times 2c-\left(-2\left(-7\right)\right)}&-\frac{-2}{3\times 2c-\left(-2\left(-7\right)\right)}\\-\frac{-7}{3\times 2c-\left(-2\left(-7\right)\right)}&\frac{3}{3\times 2c-\left(-2\left(-7\right)\right)}\end{matrix}\right)\left(\begin{matrix}-6\\-5\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{c}{3c-7}&\frac{1}{3c-7}\\\frac{7}{2\left(3c-7\right)}&\frac{3}{2\left(3c-7\right)}\end{matrix}\right)\left(\begin{matrix}-6\\-5\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{c}{3c-7}\left(-6\right)+\frac{1}{3c-7}\left(-5\right)\\\frac{7}{2\left(3c-7\right)}\left(-6\right)+\frac{3}{2\left(3c-7\right)}\left(-5\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{6c+5}{3c-7}\\-\frac{57}{2\left(3c-7\right)}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{6c+5}{3c-7},y=-\frac{57}{2\left(3c-7\right)}$$

Ngā Upane e Whakamahi i te Tangohanga

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

$$3x+6=2y$$

Tangohia te 2y mai i ngā taha e rua.

$$3x+6-2y=0$$

Tangohia te 6 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$3x-2y=-6$$

Whakaarohia te whārite tuarua. Tangohia te 7x mai i ngā taha e rua.

$$2cy+5-7x=0$$

Tangohia te 5 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$2cy-7x=-5$$

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.

$$3x-2y=-6,-7x+2cy=-5$$

Kia ōrite ai a 3x me -7x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -7 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$-7\times 3x-7\left(-2\right)y=-7\left(-6\right),3\left(-7\right)x+3\times 2cy=3\left(-5\right)$$

Whakarūnātia.

$$-21x+14y=42,-21x+6cy=-15$$

Me tango -21x+6cy=-15 mai i -21x+14y=42 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-21x+21x+14y+\left(-6c\right)y=42+15$$

Tāpiri -21x ki te 21x. Ka whakakore atu ngā kupu -21x me 21x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$14y+\left(-6c\right)y=42+15$$

Tāpiri 14y ki te -6cy.

$$\left(14-6c\right)y=42+15$$

Tāpiri 42 ki te 15.

$$\left(14-6c\right)y=57$$

Whakawehea ngā taha e rua ki te 14-6c.

$$y=\frac{57}{2\left(7-3c\right)}$$

Whakaurua te \frac{57}{2\left(7-3c\right)} mō y ki -7x+2cy=-5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-7x+2c\times \frac{57}{2\left(7-3c\right)}=-5$$

Whakareatia 2c ki te \frac{57}{2\left(7-3c\right)}.

$$-7x+\frac{57c}{7-3c}=-5$$

Me tango \frac{57c}{7-3c} mai i ngā taha e rua o te whārite.

$$-7x=-\frac{7\left(6c+5\right)}{7-3c}$$

Whakawehea ngā taha e rua ki te -7.

$$x=\frac{6c+5}{7-3c}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{6c+5}{7-3c},y=\frac{57}{2\left(7-3c\right)}$$