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

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

Atrast x

$$x=y+2$$

Atrisinājuma darbības

Kāpiniet kvadrātā abas vienādojuma puses.

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

Lietojiet Ņūtona binomu \left(a-b\right)^{2}=a^{2}-2ab+b^{2}, lai izvērstu \left(7-x\right)^{2}.

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

Lietojiet Ņūtona binomu \left(a-b\right)^{2}=a^{2}-2ab+b^{2}, lai izvērstu \left(1-y\right)^{2}.

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

Saskaitiet 49 un 1, lai iegūtu 50.

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

Aprēķiniet \sqrt{50-14x+x^{2}-2y+y^{2}} pakāpē 2 un iegūstiet 50-14x+x^{2}-2y+y^{2}.

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

Lietojiet Ņūtona binomu \left(a-b\right)^{2}=a^{2}-2ab+b^{2}, lai izvērstu \left(3-x\right)^{2}.

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

Lietojiet Ņūtona binomu \left(a-b\right)^{2}=a^{2}-2ab+b^{2}, lai izvērstu \left(5-y\right)^{2}.

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

Saskaitiet 9 un 25, lai iegūtu 34.

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

Aprēķiniet \sqrt{34-6x+x^{2}-10y+y^{2}} pakāpē 2 un iegūstiet 34-6x+x^{2}-10y+y^{2}.

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

Pievienot 6x abās pusēs.

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

Savelciet -14x un 6x, lai iegūtu -8x.

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

Atņemiet x^{2} no abām pusēm.

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

Savelciet x^{2} un -x^{2}, lai iegūtu 0.

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

Atņemiet 50 no abām pusēm.

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

Atņemiet 50 no 34, lai iegūtu -16.

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

Pievienot 2y abās pusēs.

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

Savelciet -10y un 2y, lai iegūtu -8y.

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

Atņemiet y^{2} no abām pusēm.

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

Savelciet y^{2} un -y^{2}, lai iegūtu 0.

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

Vienādojums ir standarta formā.

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

Daliet abas puses ar -8.

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

Dalīšana ar -8 atsauc reizināšanu ar -8.

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

Daliet -16-8y ar -8.

$$x=y+2$$

Ar y+2 aizvietojiet x vienādojumā \sqrt{\left(7-x\right)^{2}+\left(1-y\right)^{2}}=\sqrt{\left(3-x\right)^{2}+\left(5-y\right)^{2}}.

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

Vienkāršojiet. Vērtība x=y+2 atbilst vienādojumam.

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

Vienādojumam \sqrt{\left(7-x\right)^{2}+\left(1-y\right)^{2}}=\sqrt{\left(3-x\right)^{2}+\left(5-y\right)^{2}} ir unikāls risinājums.

$$x=y+2$$

Atrast y

$$y=x-2$$

Atrisinājuma darbības

Kāpiniet kvadrātā abas vienādojuma puses.

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

Lietojiet Ņūtona binomu \left(a-b\right)^{2}=a^{2}-2ab+b^{2}, lai izvērstu \left(7-x\right)^{2}.

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

Lietojiet Ņūtona binomu \left(a-b\right)^{2}=a^{2}-2ab+b^{2}, lai izvērstu \left(1-y\right)^{2}.

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

Saskaitiet 49 un 1, lai iegūtu 50.

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

Aprēķiniet \sqrt{50-14x+x^{2}-2y+y^{2}} pakāpē 2 un iegūstiet 50-14x+x^{2}-2y+y^{2}.

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

Lietojiet Ņūtona binomu \left(a-b\right)^{2}=a^{2}-2ab+b^{2}, lai izvērstu \left(3-x\right)^{2}.

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

Lietojiet Ņūtona binomu \left(a-b\right)^{2}=a^{2}-2ab+b^{2}, lai izvērstu \left(5-y\right)^{2}.

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

Saskaitiet 9 un 25, lai iegūtu 34.

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

Aprēķiniet \sqrt{34-6x+x^{2}-10y+y^{2}} pakāpē 2 un iegūstiet 34-6x+x^{2}-10y+y^{2}.

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

Pievienot 10y abās pusēs.

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

Savelciet -2y un 10y, lai iegūtu 8y.

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

Atņemiet y^{2} no abām pusēm.

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

Savelciet y^{2} un -y^{2}, lai iegūtu 0.

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

Atņemiet 50 no abām pusēm.

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

Atņemiet 50 no 34, lai iegūtu -16.

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

Pievienot 14x abās pusēs.

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

Savelciet -6x un 14x, lai iegūtu 8x.

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

Atņemiet x^{2} no abām pusēm.

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

Savelciet x^{2} un -x^{2}, lai iegūtu 0.

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

Vienādojums ir standarta formā.

$$8y=8x-16$$

Daliet abas puses ar 8.

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

Dalīšana ar 8 atsauc reizināšanu ar 8.

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

Daliet -16+8x ar 8.

$$y=x-2$$

Ar x-2 aizvietojiet y vienādojumā \sqrt{\left(7-x\right)^{2}+\left(1-y\right)^{2}}=\sqrt{\left(3-x\right)^{2}+\left(5-y\right)^{2}}.

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

Vienkāršojiet. Vērtība y=x-2 atbilst vienādojumam.

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

Vienādojumam \sqrt{\left(7-x\right)^{2}+\left(1-y\right)^{2}}=\sqrt{\left(3-x\right)^{2}+\left(5-y\right)^{2}} ir unikāls risinājums.

$$y=x-2$$