Whakaoti i te {l}{x+2/y+5=x+7/y}{2x-4y=-1}

$$\left\{ \begin{array} { l } { \frac { x + 2 } { y + 5 } = \frac { x + 7 } { y } } \\ { 2 x - 4 y = - 1 } \end{array} \right.$$

Whakaoti mō x, y

$$x = -\frac{29}{6} = -4\frac{5}{6} \approx -4.833333333$$

$$y = -\frac{13}{6} = -2\frac{1}{6} \approx -2.166666667$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tē taea kia ōrite te tāupe y ki tētahi o ngā uara -5,0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te y\left(y+5\right), arā, te tauraro pātahi he tino iti rawa te kitea o y+5,y.

$$y\left(x+2\right)=\left(y+5\right)\left(x+7\right)$$

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

$$yx+2y=\left(y+5\right)\left(x+7\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te y+5 ki te x+7.

$$yx+2y=yx+7y+5x+35$$

Tangohia te yx mai i ngā taha e rua.

$$yx+2y-yx=7y+5x+35$$

Pahekotia te yx me -yx, ka 0.

$$2y=7y+5x+35$$

Tangohia te 7y mai i ngā taha e rua.

$$2y-7y=5x+35$$

Pahekotia te 2y me -7y, ka -5y.

$$-5y=5x+35$$

Whakawehea ngā taha e rua ki te -5.

$$y=-\frac{1}{5}\left(5x+35\right)$$

Whakareatia -\frac{1}{5} ki te 35+5x.

$$y=-x-7$$

Whakakapia te -x-7 mō te y ki tērā atu whārite, -4y+2x=-1.

$$-4\left(-x-7\right)+2x=-1$$

Whakareatia -4 ki te -x-7.

$$4x+28+2x=-1$$

Tāpiri 4x ki te 2x.

$$6x+28=-1$$

Me tango 28 mai i ngā taha e rua o te whārite.

$$6x=-29$$

Whakawehea ngā taha e rua ki te 6.

$$x=-\frac{29}{6}$$

Whakaurua te -\frac{29}{6} mō x ki y=-x-7. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=-\left(-\frac{29}{6}\right)-7$$

Whakareatia -1 ki te -\frac{29}{6}.

$$y=\frac{29}{6}-7$$

Tāpiri -7 ki te \frac{29}{6}.

$$y=-\frac{13}{6}$$

Kua oti te pūnaha te whakatau.

$$y=-\frac{13}{6},x=-\frac{29}{6}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tē taea kia ōrite te tāupe y ki tētahi o ngā uara -5,0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te y\left(y+5\right), arā, te tauraro pātahi he tino iti rawa te kitea o y+5,y.

$$y\left(x+2\right)=\left(y+5\right)\left(x+7\right)$$

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

$$yx+2y=\left(y+5\right)\left(x+7\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te y+5 ki te x+7.

$$yx+2y=yx+7y+5x+35$$

Tangohia te yx mai i ngā taha e rua.

$$yx+2y-yx=7y+5x+35$$

Pahekotia te yx me -yx, ka 0.

$$2y=7y+5x+35$$

Tangohia te 7y mai i ngā taha e rua.

$$2y-7y=5x+35$$

Pahekotia te 2y me -7y, ka -5y.

$$-5y=5x+35$$

Tangohia te 5x mai i ngā taha e rua.

$$-5y-5x=35$$

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

$$-5y-5x=35,-4y+2x=-1$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-5&-5\\-4&2\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}35\\-1\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-5&-5\\-4&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}-5&-5\\-4&2\end{matrix}\right))\left(\begin{matrix}-5&-5\\-4&2\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}-5&-5\\-4&2\end{matrix}\right))\left(\begin{matrix}35\\-1\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}-5&-5\\-4&2\end{matrix}\right))\left(\begin{matrix}35\\-1\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}-5&-5\\-4&2\end{matrix}\right))\left(\begin{matrix}35\\-1\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{2}{-5\times 2-\left(-5\left(-4\right)\right)}&-\frac{-5}{-5\times 2-\left(-5\left(-4\right)\right)}\\-\frac{-4}{-5\times 2-\left(-5\left(-4\right)\right)}&-\frac{5}{-5\times 2-\left(-5\left(-4\right)\right)}\end{matrix}\right)\left(\begin{matrix}35\\-1\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{15}&-\frac{1}{6}\\-\frac{2}{15}&\frac{1}{6}\end{matrix}\right)\left(\begin{matrix}35\\-1\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{15}\times 35-\frac{1}{6}\left(-1\right)\\-\frac{2}{15}\times 35+\frac{1}{6}\left(-1\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{13}{6}\\-\frac{29}{6}\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=-\frac{13}{6},x=-\frac{29}{6}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Tē taea kia ōrite te tāupe y ki tētahi o ngā uara -5,0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te y\left(y+5\right), arā, te tauraro pātahi he tino iti rawa te kitea o y+5,y.

$$y\left(x+2\right)=\left(y+5\right)\left(x+7\right)$$

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

$$yx+2y=\left(y+5\right)\left(x+7\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te y+5 ki te x+7.

$$yx+2y=yx+7y+5x+35$$

Tangohia te yx mai i ngā taha e rua.

$$yx+2y-yx=7y+5x+35$$

Pahekotia te yx me -yx, ka 0.

$$2y=7y+5x+35$$

Tangohia te 7y mai i ngā taha e rua.

$$2y-7y=5x+35$$

Pahekotia te 2y me -7y, ka -5y.

$$-5y=5x+35$$

Tangohia te 5x mai i ngā taha e rua.

$$-5y-5x=35$$

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.

$$-5y-5x=35,-4y+2x=-1$$

Kia ōrite ai a -5y me -4y, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -4 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te -5.

$$-4\left(-5\right)y-4\left(-5\right)x=-4\times 35,-5\left(-4\right)y-5\times 2x=-5\left(-1\right)$$

Whakarūnātia.

$$20y+20x=-140,20y-10x=5$$

Me tango 20y-10x=5 mai i 20y+20x=-140 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$20y-20y+20x+10x=-140-5$$

Tāpiri 20y ki te -20y. Ka whakakore atu ngā kupu 20y me -20y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$20x+10x=-140-5$$

Tāpiri 20x ki te 10x.

$$30x=-140-5$$

Tāpiri -140 ki te -5.

$$30x=-145$$

Whakawehea ngā taha e rua ki te 30.

$$x=-\frac{29}{6}$$

Whakaurua te -\frac{29}{6} mō x ki -4y+2x=-1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$-4y+2\left(-\frac{29}{6}\right)=-1$$

Whakareatia 2 ki te -\frac{29}{6}.

$$-4y-\frac{29}{3}=-1$$

Me tāpiri \frac{29}{3} ki ngā taha e rua o te whārite.

$$-4y=\frac{26}{3}$$

Whakawehea ngā taha e rua ki te -4.

$$y=-\frac{13}{6}$$

Kua oti te pūnaha te whakatau.

$$y=-\frac{13}{6},x=-\frac{29}{6}$$