Issolvi sqrt{(7-x)^2+(1-y)^2}=sqrt{(3-x)^2+(5-y)^2}

$$\sqrt { ( 7 - x ) ^ { 2 } + ( 1 - y ) ^ { 2 } } = \sqrt { ( 3 - x ) ^ { 2 } + ( 5 - y ) ^ { 2 } }$$

Solvi għal x

$$x=y+2$$

Passi tas-Soluzzjonijiet

Ikkwadra ż-żewġ naħat tal-ekwazzjoni.

$$\left(\sqrt{\left(7-x\right)^{2}+\left(1-y\right)^{2}}\right)^{2}=\left(\sqrt{\left(3-x\right)^{2}+\left(5-y\right)^{2}}\right)^{2}$$

Uża teorema binomjali \left(a-b\right)^{2}=a^{2}-2ab+b^{2} biex tespandi \left(7-x\right)^{2}.

$$\left(\sqrt{49-14x+x^{2}+\left(1-y\right)^{2}}\right)^{2}=\left(\sqrt{\left(3-x\right)^{2}+\left(5-y\right)^{2}}\right)^{2}$$

Uża teorema binomjali \left(a-b\right)^{2}=a^{2}-2ab+b^{2} biex tespandi \left(1-y\right)^{2}.

$$\left(\sqrt{49-14x+x^{2}+1-2y+y^{2}}\right)^{2}=\left(\sqrt{\left(3-x\right)^{2}+\left(5-y\right)^{2}}\right)^{2}$$

Żid 49 u 1 biex tikseb 50.

$$\left(\sqrt{50-14x+x^{2}-2y+y^{2}}\right)^{2}=\left(\sqrt{\left(3-x\right)^{2}+\left(5-y\right)^{2}}\right)^{2}$$

Ikkalkula \sqrt{50-14x+x^{2}-2y+y^{2}} bil-power ta' 2 u tikseb 50-14x+x^{2}-2y+y^{2}.

$$50-14x+x^{2}-2y+y^{2}=\left(\sqrt{\left(3-x\right)^{2}+\left(5-y\right)^{2}}\right)^{2}$$

Uża teorema binomjali \left(a-b\right)^{2}=a^{2}-2ab+b^{2} biex tespandi \left(3-x\right)^{2}.

$$50-14x+x^{2}-2y+y^{2}=\left(\sqrt{9-6x+x^{2}+\left(5-y\right)^{2}}\right)^{2}$$

Uża teorema binomjali \left(a-b\right)^{2}=a^{2}-2ab+b^{2} biex tespandi \left(5-y\right)^{2}.

$$50-14x+x^{2}-2y+y^{2}=\left(\sqrt{9-6x+x^{2}+25-10y+y^{2}}\right)^{2}$$

Żid 9 u 25 biex tikseb 34.

$$50-14x+x^{2}-2y+y^{2}=\left(\sqrt{34-6x+x^{2}-10y+y^{2}}\right)^{2}$$

Ikkalkula \sqrt{34-6x+x^{2}-10y+y^{2}} bil-power ta' 2 u tikseb 34-6x+x^{2}-10y+y^{2}.

$$50-14x+x^{2}-2y+y^{2}=34-6x+x^{2}-10y+y^{2}$$

Żid 6x maż-żewġ naħat.

$$50-14x+x^{2}-2y+y^{2}+6x=34+x^{2}-10y+y^{2}$$

Ikkombina -14x u 6x biex tikseb -8x.

$$50-8x+x^{2}-2y+y^{2}=34+x^{2}-10y+y^{2}$$

Naqqas x^{2} miż-żewġ naħat.

$$50-8x+x^{2}-2y+y^{2}-x^{2}=34-10y+y^{2}$$

Ikkombina x^{2} u -x^{2} biex tikseb 0.

$$50-8x-2y+y^{2}=34-10y+y^{2}$$

Naqqas 50 miż-żewġ naħat.

$$-8x-2y+y^{2}=34-10y+y^{2}-50$$

Naqqas 50 minn 34 biex tikseb -16.

$$-8x-2y+y^{2}=-16-10y+y^{2}$$

Żid 2y maż-żewġ naħat.

$$-8x+y^{2}=-16-10y+y^{2}+2y$$

Ikkombina -10y u 2y biex tikseb -8y.

$$-8x+y^{2}=-16-8y+y^{2}$$

Naqqas y^{2} miż-żewġ naħat.

$$-8x=-16-8y+y^{2}-y^{2}$$

Ikkombina y^{2} u -y^{2} biex tikseb 0.

$$-8x=-16-8y$$

L-ekwazzjoni hija f'forma standard.

$$-8x=-8y-16$$

Iddividi ż-żewġ naħat b'-8.

$$\frac{-8x}{-8}=\frac{-8y-16}{-8}$$

Meta tiddividi b'-8 titneħħa l-multiplikazzjoni b'-8.

$$x=\frac{-8y-16}{-8}$$

Iddividi -16-8y b'-8.

$$x=y+2$$

Issostitwixxi y+2 għal x fl-ekwazzjoni l-oħra \sqrt{\left(7-x\right)^{2}+\left(1-y\right)^{2}}=\sqrt{\left(3-x\right)^{2}+\left(5-y\right)^{2}}.

$$\sqrt{\left(7-\left(y+2\right)\right)^{2}+\left(1-y\right)^{2}}=\sqrt{\left(3-\left(y+2\right)\right)^{2}+\left(5-y\right)^{2}}$$

Issimplifika. Il-valur x=y+2 jissodisfa l-ekwazzjoni.

$$\left(2y^{2}-12y+26\right)^{\frac{1}{2}}=\left(2y^{2}-12y+26\right)^{\frac{1}{2}}$$

Ekwazzjoni \sqrt{\left(7-x\right)^{2}+\left(1-y\right)^{2}}=\sqrt{\left(3-x\right)^{2}+\left(5-y\right)^{2}} għandha soluzzjoni unika.

$$x=y+2$$

Solvi għal y

$$y=x-2$$

Passi tas-Soluzzjonijiet

Ikkwadra ż-żewġ naħat tal-ekwazzjoni.

$$\left(\sqrt{\left(7-x\right)^{2}+\left(1-y\right)^{2}}\right)^{2}=\left(\sqrt{\left(3-x\right)^{2}+\left(5-y\right)^{2}}\right)^{2}$$

Uża teorema binomjali \left(a-b\right)^{2}=a^{2}-2ab+b^{2} biex tespandi \left(7-x\right)^{2}.

$$\left(\sqrt{49-14x+x^{2}+\left(1-y\right)^{2}}\right)^{2}=\left(\sqrt{\left(3-x\right)^{2}+\left(5-y\right)^{2}}\right)^{2}$$

Uża teorema binomjali \left(a-b\right)^{2}=a^{2}-2ab+b^{2} biex tespandi \left(1-y\right)^{2}.

$$\left(\sqrt{49-14x+x^{2}+1-2y+y^{2}}\right)^{2}=\left(\sqrt{\left(3-x\right)^{2}+\left(5-y\right)^{2}}\right)^{2}$$

Żid 49 u 1 biex tikseb 50.

$$\left(\sqrt{50-14x+x^{2}-2y+y^{2}}\right)^{2}=\left(\sqrt{\left(3-x\right)^{2}+\left(5-y\right)^{2}}\right)^{2}$$

Ikkalkula \sqrt{50-14x+x^{2}-2y+y^{2}} bil-power ta' 2 u tikseb 50-14x+x^{2}-2y+y^{2}.

$$50-14x+x^{2}-2y+y^{2}=\left(\sqrt{\left(3-x\right)^{2}+\left(5-y\right)^{2}}\right)^{2}$$

Uża teorema binomjali \left(a-b\right)^{2}=a^{2}-2ab+b^{2} biex tespandi \left(3-x\right)^{2}.

$$50-14x+x^{2}-2y+y^{2}=\left(\sqrt{9-6x+x^{2}+\left(5-y\right)^{2}}\right)^{2}$$

Uża teorema binomjali \left(a-b\right)^{2}=a^{2}-2ab+b^{2} biex tespandi \left(5-y\right)^{2}.

$$50-14x+x^{2}-2y+y^{2}=\left(\sqrt{9-6x+x^{2}+25-10y+y^{2}}\right)^{2}$$

Żid 9 u 25 biex tikseb 34.

$$50-14x+x^{2}-2y+y^{2}=\left(\sqrt{34-6x+x^{2}-10y+y^{2}}\right)^{2}$$

Ikkalkula \sqrt{34-6x+x^{2}-10y+y^{2}} bil-power ta' 2 u tikseb 34-6x+x^{2}-10y+y^{2}.

$$50-14x+x^{2}-2y+y^{2}=34-6x+x^{2}-10y+y^{2}$$

Żid 10y maż-żewġ naħat.

$$50-14x+x^{2}-2y+y^{2}+10y=34-6x+x^{2}+y^{2}$$

Ikkombina -2y u 10y biex tikseb 8y.

$$50-14x+x^{2}+8y+y^{2}=34-6x+x^{2}+y^{2}$$

Naqqas y^{2} miż-żewġ naħat.

$$50-14x+x^{2}+8y+y^{2}-y^{2}=34-6x+x^{2}$$

Ikkombina y^{2} u -y^{2} biex tikseb 0.

$$50-14x+x^{2}+8y=34-6x+x^{2}$$

Naqqas 50 miż-żewġ naħat.

$$-14x+x^{2}+8y=34-6x+x^{2}-50$$

Naqqas 50 minn 34 biex tikseb -16.

$$-14x+x^{2}+8y=-16-6x+x^{2}$$

Żid 14x maż-żewġ naħat.

$$x^{2}+8y=-16-6x+x^{2}+14x$$

Ikkombina -6x u 14x biex tikseb 8x.

$$x^{2}+8y=-16+8x+x^{2}$$

Naqqas x^{2} miż-żewġ naħat.

$$8y=-16+8x+x^{2}-x^{2}$$

Ikkombina x^{2} u -x^{2} biex tikseb 0.

$$8y=-16+8x$$

L-ekwazzjoni hija f'forma standard.

$$8y=8x-16$$

Iddividi ż-żewġ naħat b'8.

$$\frac{8y}{8}=\frac{8x-16}{8}$$

Meta tiddividi b'8 titneħħa l-multiplikazzjoni b'8.

$$y=\frac{8x-16}{8}$$

Iddividi -16+8x b'8.

$$y=x-2$$

Issostitwixxi x-2 għal y fl-ekwazzjoni l-oħra \sqrt{\left(7-x\right)^{2}+\left(1-y\right)^{2}}=\sqrt{\left(3-x\right)^{2}+\left(5-y\right)^{2}}.

$$\sqrt{\left(7-x\right)^{2}+\left(1-\left(x-2\right)\right)^{2}}=\sqrt{\left(3-x\right)^{2}+\left(5-\left(x-2\right)\right)^{2}}$$

Issimplifika. Il-valur y=x-2 jissodisfa l-ekwazzjoni.

$$\left(2x^{2}-20x+58\right)^{\frac{1}{2}}=\left(2x^{2}-20x+58\right)^{\frac{1}{2}}$$

Ekwazzjoni \sqrt{\left(7-x\right)^{2}+\left(1-y\right)^{2}}=\sqrt{\left(3-x\right)^{2}+\left(5-y\right)^{2}} għandha soluzzjoni unika.

$$y=x-2$$