Whakaoti i te {l}{x^2-y^2=7}{2y=2+x}

$$\left. \begin{array} { l } { x ^ { 2 } - y ^ { 2 } = 7 } \\ { 2 y = 2 + x } \end{array} \right.$$

Whakaoti mō x, y

$$x=4\text{, }y=3$$

$$x=-\frac{8}{3}\approx -2.666666667\text{, }y=-\frac{1}{3}\approx -0.333333333$$

Ngā Upane e Whakamahi ana i te Whakakapinga me te Tikanga Tātai Pūrua

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

$$2y-x=2$$

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.

$$2y-x=2,x^{2}-y^{2}=7$$

Whakaotia te 2y-x=2 mō y mā te wehe i te y i te taha mauī o te tohu ōrite.

$$2y-x=2$$

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

$$2y=x+2$$

Whakawehea ngā taha e rua ki te 2.

$$y=\frac{1}{2}x+1$$

Whakakapia te \frac{1}{2}x+1 mō te y ki tērā atu whārite, x^{2}-y^{2}=7.

$$x^{2}-\left(\frac{1}{2}x+1\right)^{2}=7$$

Pūrua \frac{1}{2}x+1.

$$x^{2}-\left(\frac{1}{4}x^{2}+x+1\right)=7$$

Whakareatia -1 ki te \frac{1}{4}x^{2}+x+1.

$$x^{2}-\frac{1}{4}x^{2}-x-1=7$$

Tāpiri x^{2} ki te -\frac{1}{4}x^{2}.

$$\frac{3}{4}x^{2}-x-1=7$$

Me tango 7 mai i ngā taha e rua o te whārite.

$$\frac{3}{4}x^{2}-x-8=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1-\left(\frac{1}{2}\right)^{2} mō a, -\frac{1}{2}\times 2 mō b, me -8 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\times \frac{3}{4}\left(-8\right)}}{2\times \frac{3}{4}}$$

Whakareatia -4 ki te 1-\left(\frac{1}{2}\right)^{2}.

$$x=\frac{-\left(-1\right)±\sqrt{1-3\left(-8\right)}}{2\times \frac{3}{4}}$$

Whakareatia -3 ki te -8.

$$x=\frac{-\left(-1\right)±\sqrt{1+24}}{2\times \frac{3}{4}}$$

Tāpiri 1 ki te 24.

$$x=\frac{-\left(-1\right)±\sqrt{25}}{2\times \frac{3}{4}}$$

Tuhia te pūtakerua o te 25.

$$x=\frac{-\left(-1\right)±5}{2\times \frac{3}{4}}$$

Ko te tauaro o -\frac{1}{2}\times 2 ko 1.

$$x=\frac{1±5}{2\times \frac{3}{4}}$$

Whakareatia 2 ki te 1-\left(\frac{1}{2}\right)^{2}.

$$x=\frac{1±5}{\frac{3}{2}}$$

Nā, me whakaoti te whārite x=\frac{1±5}{\frac{3}{2}} ina he tāpiri te ±. Tāpiri 1 ki te 5.

$$x=\frac{6}{\frac{3}{2}}$$

Whakawehe 6 ki te \frac{3}{2} mā te whakarea 6 ki te tau huripoki o \frac{3}{2}.

$$x=4$$

Nā, me whakaoti te whārite x=\frac{1±5}{\frac{3}{2}} ina he tango te ±. Tango 5 mai i 1.

$$x=-\frac{4}{\frac{3}{2}}$$

Whakawehe -4 ki te \frac{3}{2} mā te whakarea -4 ki te tau huripoki o \frac{3}{2}.

$$x=-\frac{8}{3}$$

E rua ngā otinga mō x: 4 me -\frac{8}{3}. Me whakakapi 4 mō x ki te whārite y=\frac{1}{2}x+1 hei kimi i te otinga hāngai mō y e pai ai ki ngā whārite e rua.

$$y=\frac{1}{2}\times 4+1$$

Whakareatia \frac{1}{2} ki te 4.

$$y=2+1$$

Tāpiri \frac{1}{2}\times 4 ki te 1.

$$y=3$$

Me whakakapi te -\frac{8}{3} ināianei mō te x ki te whārite y=\frac{1}{2}x+1 ka whakaoti hei kimi i te otinga hāngai mō y e pai ai ki ngā whārite e rua.

$$y=\frac{1}{2}\left(-\frac{8}{3}\right)+1$$

Whakareatia \frac{1}{2} ki te -\frac{8}{3} 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.

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

Tāpiri -\frac{8}{3}\times \frac{1}{2} ki te 1.

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

Kua oti te pūnaha te whakatau.

$$y=3,x=4\text{ or }y=-\frac{1}{3},x=-\frac{8}{3}$$