Whakaoti i te {l}{(x-2)^2-2(x-2y)=1-(3-x)(3+x)}{2x+y=4}

$$\left\{ \begin{array} { l } { ( x - 2 ) ^ { 2 } - 2 ( x - 2 y ) = 1 - ( 3 - x ) ( 3 + x ) } \\ { 2 x + y = 4 } \end{array} \right.$$

Whakaoti mō x, y

$$x=2$$

$$y=0$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(x-2\right)^{2}.

$$x^{2}-4x+4-2\left(x-2y\right)=1-\left(3-x\right)\left(3+x\right)$$

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

$$x^{2}-4x+4-2x+4y=1-\left(3-x\right)\left(3+x\right)$$

Pahekotia te -4x me -2x, ka -6x.

$$x^{2}-6x+4+4y=1-\left(3-x\right)\left(3+x\right)$$

Whakaarohia te \left(3-x\right)\left(3+x\right). Ka taea te whakareanga te panoni ki te rerekētanga o ngā pūrua mā te ture: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Pūrua 3.

$$x^{2}-6x+4+4y=1-\left(9-x^{2}\right)$$

Hei kimi i te tauaro o 9-x^{2}, kimihia te tauaro o ia taurangi.

$$x^{2}-6x+4+4y=1-9+x^{2}$$

Tangohia te 9 i te 1, ka -8.

$$x^{2}-6x+4+4y=-8+x^{2}$$

Tangohia te x^{2} mai i ngā taha e rua.

$$x^{2}-6x+4+4y-x^{2}=-8$$

Pahekotia te x^{2} me -x^{2}, ka 0.

$$-6x+4+4y=-8$$

Tangohia te 4 mai i ngā taha e rua.

$$-6x+4y=-8-4$$

Tangohia te 4 i te -8, ka -12.

$$-6x+4y=-12$$

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.

$$-6x+4y=-12,2x+y=4$$

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

$$-6x+4y=-12$$

Me tango 4y mai i ngā taha e rua o te whārite.

$$-6x=-4y-12$$

Whakawehea ngā taha e rua ki te -6.

$$x=-\frac{1}{6}\left(-4y-12\right)$$

Whakareatia -\frac{1}{6} ki te -4y-12.

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

Whakakapia te \frac{2y}{3}+2 mō te x ki tērā atu whārite, 2x+y=4.

$$2\left(\frac{2}{3}y+2\right)+y=4$$

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

$$\frac{4}{3}y+4+y=4$$

Tāpiri \frac{4y}{3} ki te y.

$$\frac{7}{3}y+4=4$$

Me tango 4 mai i ngā taha e rua o te whārite.

$$\frac{7}{3}y=0$$

Whakawehea ngā taha e rua o te whārite ki te \frac{7}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=0$$

Whakaurua te 0 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.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=0$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(x-2\right)^{2}.

$$x^{2}-4x+4-2\left(x-2y\right)=1-\left(3-x\right)\left(3+x\right)$$

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

$$x^{2}-4x+4-2x+4y=1-\left(3-x\right)\left(3+x\right)$$

Pahekotia te -4x me -2x, ka -6x.

$$x^{2}-6x+4+4y=1-\left(3-x\right)\left(3+x\right)$$

Whakaarohia te \left(3-x\right)\left(3+x\right). Ka taea te whakareanga te panoni ki te rerekētanga o ngā pūrua mā te ture: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Pūrua 3.

$$x^{2}-6x+4+4y=1-\left(9-x^{2}\right)$$

Hei kimi i te tauaro o 9-x^{2}, kimihia te tauaro o ia taurangi.

$$x^{2}-6x+4+4y=1-9+x^{2}$$

Tangohia te 9 i te 1, ka -8.

$$x^{2}-6x+4+4y=-8+x^{2}$$

Tangohia te x^{2} mai i ngā taha e rua.

$$x^{2}-6x+4+4y-x^{2}=-8$$

Pahekotia te x^{2} me -x^{2}, ka 0.

$$-6x+4+4y=-8$$

Tangohia te 4 mai i ngā taha e rua.

$$-6x+4y=-8-4$$

Tangohia te 4 i te -8, ka -12.

$$-6x+4y=-12$$

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

$$-6x+4y=-12,2x+y=4$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-6&4\\2&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-12\\4\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-6&4\\2&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}-6&4\\2&1\end{matrix}\right))\left(\begin{matrix}-6&4\\2&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-6&4\\2&1\end{matrix}\right))\left(\begin{matrix}-12\\4\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-6&4\\2&1\end{matrix}\right))\left(\begin{matrix}-12\\4\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-6&4\\2&1\end{matrix}\right))\left(\begin{matrix}-12\\4\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{-6-4\times 2}&-\frac{4}{-6-4\times 2}\\-\frac{2}{-6-4\times 2}&-\frac{6}{-6-4\times 2}\end{matrix}\right)\left(\begin{matrix}-12\\4\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{14}&\frac{2}{7}\\\frac{1}{7}&\frac{3}{7}\end{matrix}\right)\left(\begin{matrix}-12\\4\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{14}\left(-12\right)+\frac{2}{7}\times 4\\\frac{1}{7}\left(-12\right)+\frac{3}{7}\times 4\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2\\0\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=2,y=0$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(x-2\right)^{2}.

$$x^{2}-4x+4-2\left(x-2y\right)=1-\left(3-x\right)\left(3+x\right)$$

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

$$x^{2}-4x+4-2x+4y=1-\left(3-x\right)\left(3+x\right)$$

Pahekotia te -4x me -2x, ka -6x.

$$x^{2}-6x+4+4y=1-\left(3-x\right)\left(3+x\right)$$

Whakaarohia te \left(3-x\right)\left(3+x\right). Ka taea te whakareanga te panoni ki te rerekētanga o ngā pūrua mā te ture: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Pūrua 3.

$$x^{2}-6x+4+4y=1-\left(9-x^{2}\right)$$

Hei kimi i te tauaro o 9-x^{2}, kimihia te tauaro o ia taurangi.

$$x^{2}-6x+4+4y=1-9+x^{2}$$

Tangohia te 9 i te 1, ka -8.

$$x^{2}-6x+4+4y=-8+x^{2}$$

Tangohia te x^{2} mai i ngā taha e rua.

$$x^{2}-6x+4+4y-x^{2}=-8$$

Pahekotia te x^{2} me -x^{2}, ka 0.

$$-6x+4+4y=-8$$

Tangohia te 4 mai i ngā taha e rua.

$$-6x+4y=-8-4$$

Tangohia te 4 i te -8, ka -12.

$$-6x+4y=-12$$

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.

$$-6x+4y=-12,2x+y=4$$

Kia ōrite ai a -6x me 2x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te -6.

$$2\left(-6\right)x+2\times 4y=2\left(-12\right),-6\times 2x-6y=-6\times 4$$

Whakarūnātia.

$$-12x+8y=-24,-12x-6y=-24$$

Me tango -12x-6y=-24 mai i -12x+8y=-24 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-12x+12x+8y+6y=-24+24$$

Tāpiri -12x ki te 12x. Ka whakakore atu ngā kupu -12x me 12x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$8y+6y=-24+24$$

Tāpiri 8y ki te 6y.

$$14y=-24+24$$

Tāpiri -24 ki te 24.

$$14y=0$$

Whakawehea ngā taha e rua ki te 14.

$$y=0$$

Whakaurua te 0 mō y ki 2x+y=4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x=4$$

Whakawehea ngā taha e rua ki te 2.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=0$$