Whakaoti i te {l}{-5*(x-y)+3y=2x}{2y-(6x+7)=-2}

$$\left. \begin{array} { l } { - 5 \cdot ( x - y ) + 3 y = 2 x } \\ { 2 y - ( 6 x + 7 ) = - 2 } \end{array} \right.$$

Whakaoti mō x, y

$$x = -\frac{20}{17} = -1\frac{3}{17} \approx -1.176470588$$

$$y = -\frac{35}{34} = -1\frac{1}{34} \approx -1.029411765$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te -5 ki te x-y.

$$-5x+5y+3y=2x$$

Pahekotia te 5y me 3y, ka 8y.

$$-5x+8y=2x$$

Tangohia te 2x mai i ngā taha e rua.

$$-5x+8y-2x=0$$

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

$$-7x+8y=0$$

Whakaarohia te whārite tuarua. Hei kimi i te tauaro o 6x+7, kimihia te tauaro o ia taurangi.

$$2y-6x-7=-2$$

Me tāpiri te 7 ki ngā taha e rua.

$$2y-6x=-2+7$$

Tāpirihia te -2 ki te 7, ka 5.

$$2y-6x=5$$

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.

$$-7x+8y=0,-6x+2y=5$$

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

$$-7x+8y=0$$

Me tango 8y mai i ngā taha e rua o te whārite.

$$-7x=-8y$$

Whakawehea ngā taha e rua ki te -7.

$$x=-\frac{1}{7}\left(-8\right)y$$

Whakareatia -\frac{1}{7} ki te -8y.

$$x=\frac{8}{7}y$$

Whakakapia te \frac{8y}{7} mō te x ki tērā atu whārite, -6x+2y=5.

$$-6\times \frac{8}{7}y+2y=5$$

Whakareatia -6 ki te \frac{8y}{7}.

$$-\frac{48}{7}y+2y=5$$

Tāpiri -\frac{48y}{7} ki te 2y.

$$-\frac{34}{7}y=5$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{34}{7}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-\frac{35}{34}$$

Whakaurua te -\frac{35}{34} mō y ki x=\frac{8}{7}y. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{8}{7}\left(-\frac{35}{34}\right)$$

Whakareatia \frac{8}{7} ki te -\frac{35}{34} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=-\frac{20}{17}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{20}{17},y=-\frac{35}{34}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te -5 ki te x-y.

$$-5x+5y+3y=2x$$

Pahekotia te 5y me 3y, ka 8y.

$$-5x+8y=2x$$

Tangohia te 2x mai i ngā taha e rua.

$$-5x+8y-2x=0$$

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

$$-7x+8y=0$$

Whakaarohia te whārite tuarua. Hei kimi i te tauaro o 6x+7, kimihia te tauaro o ia taurangi.

$$2y-6x-7=-2$$

Me tāpiri te 7 ki ngā taha e rua.

$$2y-6x=-2+7$$

Tāpirihia te -2 ki te 7, ka 5.

$$2y-6x=5$$

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

$$-7x+8y=0,-6x+2y=5$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-7&8\\-6&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}0\\5\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-7&8\\-6&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}-7&8\\-6&2\end{matrix}\right))\left(\begin{matrix}-7&8\\-6&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-7&8\\-6&2\end{matrix}\right))\left(\begin{matrix}0\\5\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-7&8\\-6&2\end{matrix}\right))\left(\begin{matrix}0\\5\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-7&8\\-6&2\end{matrix}\right))\left(\begin{matrix}0\\5\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{-7\times 2-8\left(-6\right)}&-\frac{8}{-7\times 2-8\left(-6\right)}\\-\frac{-6}{-7\times 2-8\left(-6\right)}&-\frac{7}{-7\times 2-8\left(-6\right)}\end{matrix}\right)\left(\begin{matrix}0\\5\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{17}&-\frac{4}{17}\\\frac{3}{17}&-\frac{7}{34}\end{matrix}\right)\left(\begin{matrix}0\\5\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{17}\times 5\\-\frac{7}{34}\times 5\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{20}{17}\\-\frac{35}{34}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{20}{17},y=-\frac{35}{34}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te -5 ki te x-y.

$$-5x+5y+3y=2x$$

Pahekotia te 5y me 3y, ka 8y.

$$-5x+8y=2x$$

Tangohia te 2x mai i ngā taha e rua.

$$-5x+8y-2x=0$$

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

$$-7x+8y=0$$

Whakaarohia te whārite tuarua. Hei kimi i te tauaro o 6x+7, kimihia te tauaro o ia taurangi.

$$2y-6x-7=-2$$

Me tāpiri te 7 ki ngā taha e rua.

$$2y-6x=-2+7$$

Tāpirihia te -2 ki te 7, ka 5.

$$2y-6x=5$$

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.

$$-7x+8y=0,-6x+2y=5$$

Kia ōrite ai a -7x me -6x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -6 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te -7.

$$-6\left(-7\right)x-6\times 8y=0,-7\left(-6\right)x-7\times 2y=-7\times 5$$

Whakarūnātia.

$$42x-48y=0,42x-14y=-35$$

Me tango 42x-14y=-35 mai i 42x-48y=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$42x-42x-48y+14y=35$$

Tāpiri 42x ki te -42x. Ka whakakore atu ngā kupu 42x me -42x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-48y+14y=35$$

Tāpiri -48y ki te 14y.

$$-34y=35$$

Whakawehea ngā taha e rua ki te -34.

$$y=-\frac{35}{34}$$

Whakaurua te -\frac{35}{34} mō y ki -6x+2y=5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-6x+2\left(-\frac{35}{34}\right)=5$$

Whakareatia 2 ki te -\frac{35}{34}.

$$-6x-\frac{35}{17}=5$$

Me tāpiri \frac{35}{17} ki ngā taha e rua o te whārite.

$$-6x=\frac{120}{17}$$

Whakawehea ngā taha e rua ki te -6.

$$x=-\frac{20}{17}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{20}{17},y=-\frac{35}{34}$$