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

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

Lahendage ja leidke x

$$x=y+2$$

Lahenduse etapid

Tõstke võrrandi mõlemad pooled ruutu.

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

Kasutage kaksliikme \left(7-x\right)^{2} arendamiseks binoomvalemit \left(a-b\right)^{2}=a^{2}-2ab+b^{2}.

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

Kasutage kaksliikme \left(1-y\right)^{2} arendamiseks binoomvalemit \left(a-b\right)^{2}=a^{2}-2ab+b^{2}.

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

Liitke 49 ja 1, et leida 50.

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

Arvutage 2 aste \sqrt{50-14x+x^{2}-2y+y^{2}} ja leidke 50-14x+x^{2}-2y+y^{2}.

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

Kasutage kaksliikme \left(3-x\right)^{2} arendamiseks binoomvalemit \left(a-b\right)^{2}=a^{2}-2ab+b^{2}.

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

Kasutage kaksliikme \left(5-y\right)^{2} arendamiseks binoomvalemit \left(a-b\right)^{2}=a^{2}-2ab+b^{2}.

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

Liitke 9 ja 25, et leida 34.

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

Arvutage 2 aste \sqrt{34-6x+x^{2}-10y+y^{2}} ja leidke 34-6x+x^{2}-10y+y^{2}.

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

Liitke 6x mõlemale poolele.

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

Kombineerige -14x ja 6x, et leida -8x.

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

Lahutage mõlemast poolest x^{2}.

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

Kombineerige x^{2} ja -x^{2}, et leida 0.

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

Lahutage mõlemast poolest 50.

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

Lahutage 50 väärtusest 34, et leida -16.

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

Liitke 2y mõlemale poolele.

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

Kombineerige -10y ja 2y, et leida -8y.

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

Lahutage mõlemast poolest y^{2}.

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

Kombineerige y^{2} ja -y^{2}, et leida 0.

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

Võrrand on standardkujul.

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

Jagage mõlemad pooled -8-ga.

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

-8-ga jagamine võtab -8-ga korrutamise tagasi.

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

Jagage -16-8y väärtusega -8.

$$x=y+2$$

Asendage x võrrandis \sqrt{\left(7-x\right)^{2}+\left(1-y\right)^{2}}=\sqrt{\left(3-x\right)^{2}+\left(5-y\right)^{2}} väärtusega y+2.

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

Lihtsustage. Väärtus x=y+2 vastab võrrandile.

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

Võrrandil \sqrt{\left(7-x\right)^{2}+\left(1-y\right)^{2}}=\sqrt{\left(3-x\right)^{2}+\left(5-y\right)^{2}} on ainus lahendus.

$$x=y+2$$

Lahendage ja leidke y

$$y=x-2$$

Lahenduse etapid

Tõstke võrrandi mõlemad pooled ruutu.

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

Kasutage kaksliikme \left(7-x\right)^{2} arendamiseks binoomvalemit \left(a-b\right)^{2}=a^{2}-2ab+b^{2}.

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

Kasutage kaksliikme \left(1-y\right)^{2} arendamiseks binoomvalemit \left(a-b\right)^{2}=a^{2}-2ab+b^{2}.

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

Liitke 49 ja 1, et leida 50.

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

Arvutage 2 aste \sqrt{50-14x+x^{2}-2y+y^{2}} ja leidke 50-14x+x^{2}-2y+y^{2}.

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

Kasutage kaksliikme \left(3-x\right)^{2} arendamiseks binoomvalemit \left(a-b\right)^{2}=a^{2}-2ab+b^{2}.

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

Kasutage kaksliikme \left(5-y\right)^{2} arendamiseks binoomvalemit \left(a-b\right)^{2}=a^{2}-2ab+b^{2}.

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

Liitke 9 ja 25, et leida 34.

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

Arvutage 2 aste \sqrt{34-6x+x^{2}-10y+y^{2}} ja leidke 34-6x+x^{2}-10y+y^{2}.

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

Liitke 10y mõlemale poolele.

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

Kombineerige -2y ja 10y, et leida 8y.

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

Lahutage mõlemast poolest y^{2}.

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

Kombineerige y^{2} ja -y^{2}, et leida 0.

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

Lahutage mõlemast poolest 50.

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

Lahutage 50 väärtusest 34, et leida -16.

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

Liitke 14x mõlemale poolele.

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

Kombineerige -6x ja 14x, et leida 8x.

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

Lahutage mõlemast poolest x^{2}.

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

Kombineerige x^{2} ja -x^{2}, et leida 0.

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

Võrrand on standardkujul.

$$8y=8x-16$$

Jagage mõlemad pooled 8-ga.

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

8-ga jagamine võtab 8-ga korrutamise tagasi.

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

Jagage -16+8x väärtusega 8.

$$y=x-2$$

Asendage y võrrandis \sqrt{\left(7-x\right)^{2}+\left(1-y\right)^{2}}=\sqrt{\left(3-x\right)^{2}+\left(5-y\right)^{2}} väärtusega x-2.

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

Lihtsustage. Väärtus y=x-2 vastab võrrandile.

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

Võrrandil \sqrt{\left(7-x\right)^{2}+\left(1-y\right)^{2}}=\sqrt{\left(3-x\right)^{2}+\left(5-y\right)^{2}} on ainus lahendus.

$$y=x-2$$