Whakaoti i te {l}{(x+2)^2+1=x^2+5y}{3x+y=1}

$$\left\{ \begin{array} { l } { ( x + 2 ) ^ { 2 } + 1 = x ^ { 2 } + 5 y } \\ { 3 x + y = 1 } \end{array} \right.$$

Whakaoti mō x, y

$$x=0$$

$$y=1$$

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}.

$$x^{2}+4x+4+1=x^{2}+5y$$

Tāpirihia te 4 ki te 1, ka 5.

$$x^{2}+4x+5=x^{2}+5y$$

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

$$x^{2}+4x+5-x^{2}=5y$$

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

$$4x+5=5y$$

Tangohia te 5y mai i ngā taha e rua.

$$4x+5-5y=0$$

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

$$4x-5y=-5$$

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.

$$4x-5y=-5,3x+y=1$$

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

$$4x-5y=-5$$

Me tāpiri 5y ki ngā taha e rua o te whārite.

$$4x=5y-5$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{1}{4}\left(5y-5\right)$$

Whakareatia \frac{1}{4} ki te -5+5y.

$$x=\frac{5}{4}y-\frac{5}{4}$$

Whakakapia te \frac{-5+5y}{4} mō te x ki tērā atu whārite, 3x+y=1.

$$3\left(\frac{5}{4}y-\frac{5}{4}\right)+y=1$$

Whakareatia 3 ki te \frac{-5+5y}{4}.

$$\frac{15}{4}y-\frac{15}{4}+y=1$$

Tāpiri \frac{15y}{4} ki te y.

$$\frac{19}{4}y-\frac{15}{4}=1$$

Me tāpiri \frac{15}{4} ki ngā taha e rua o te whārite.

$$\frac{19}{4}y=\frac{19}{4}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{19}{4}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=1$$

Whakaurua te 1 mō y ki x=\frac{5}{4}y-\frac{5}{4}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{5-5}{4}$$

Tāpiri -\frac{5}{4} ki te \frac{5}{4} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=0$$

Kua oti te pūnaha te whakatau.

$$x=0,y=1$$

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}.

$$x^{2}+4x+4+1=x^{2}+5y$$

Tāpirihia te 4 ki te 1, ka 5.

$$x^{2}+4x+5=x^{2}+5y$$

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

$$x^{2}+4x+5-x^{2}=5y$$

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

$$4x+5=5y$$

Tangohia te 5y mai i ngā taha e rua.

$$4x+5-5y=0$$

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

$$4x-5y=-5$$

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

$$4x-5y=-5,3x+y=1$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4&-5\\3&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&-5\\3&1\end{matrix}\right))\left(\begin{matrix}4&-5\\3&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-5\\3&1\end{matrix}\right))\left(\begin{matrix}-5\\1\end{matrix}\right)$$

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

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

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-5\\3&1\end{matrix}\right))\left(\begin{matrix}-5\\1\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te poukapa kōaro ko \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), kia tuhia anō ai te whārite poukapa hei rapanga whakarea poukapa.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{19}&\frac{5}{19}\\-\frac{3}{19}&\frac{4}{19}\end{matrix}\right)\left(\begin{matrix}-5\\1\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{19}\left(-5\right)+\frac{5}{19}\\-\frac{3}{19}\left(-5\right)+\frac{4}{19}\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}0\\1\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=0,y=1$$

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}.

$$x^{2}+4x+4+1=x^{2}+5y$$

Tāpirihia te 4 ki te 1, ka 5.

$$x^{2}+4x+5=x^{2}+5y$$

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

$$x^{2}+4x+5-x^{2}=5y$$

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

$$4x+5=5y$$

Tangohia te 5y mai i ngā taha e rua.

$$4x+5-5y=0$$

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

$$4x-5y=-5$$

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.

$$4x-5y=-5,3x+y=1$$

Kia ōrite ai a 4x me 3x, 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 4.

$$3\times 4x+3\left(-5\right)y=3\left(-5\right),4\times 3x+4y=4$$

Whakarūnātia.

$$12x-15y=-15,12x+4y=4$$

Me tango 12x+4y=4 mai i 12x-15y=-15 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$12x-12x-15y-4y=-15-4$$

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.

$$-15y-4y=-15-4$$

Tāpiri -15y ki te -4y.

$$-19y=-15-4$$

Tāpiri -15 ki te -4.

$$-19y=-19$$

Whakawehea ngā taha e rua ki te -19.

$$y=1$$

Whakaurua te 1 mō y ki 3x+y=1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x+1=1$$

Me tango 1 mai i ngā taha e rua o te whārite.

$$3x=0$$

Whakawehea ngā taha e rua ki te 3.

$$x=0$$

Kua oti te pūnaha te whakatau.

$$x=0,y=1$$