Solve (x-1)^{2}+(y-2)^{2}+(z-3)^2=(x-2)^2+(y+1)^2+(z-4)^2

$$( x - 1 ) ^ { 2 } + ( y - 2 ) ^ { 2 } + ( z - 3 ) ^ { 2 } = ( x - 2 ) ^ { 2 } + ( y + 1 ) ^ { 2 } + ( z - 4 ) ^ { 2 }$$

Solve for x

$$x=3y-z+\frac{7}{2}$$

Steps for Solving Linear Equation

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

$$x^{2}-2x+1+\left(y-2\right)^{2}+\left(z-3\right)^{2}=\left(x-2\right)^{2}+\left(y+1\right)^{2}+\left(z-4\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(y-2\right)^{2}.

$$x^{2}-2x+1+y^{2}-4y+4+\left(z-3\right)^{2}=\left(x-2\right)^{2}+\left(y+1\right)^{2}+\left(z-4\right)^{2}$$

Add 1 and 4 to get 5.

$$x^{2}-2x+5+y^{2}-4y+\left(z-3\right)^{2}=\left(x-2\right)^{2}+\left(y+1\right)^{2}+\left(z-4\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(z-3\right)^{2}.

$$x^{2}-2x+5+y^{2}-4y+z^{2}-6z+9=\left(x-2\right)^{2}+\left(y+1\right)^{2}+\left(z-4\right)^{2}$$

Add 5 and 9 to get 14.

$$x^{2}-2x+14+y^{2}-4y+z^{2}-6z=\left(x-2\right)^{2}+\left(y+1\right)^{2}+\left(z-4\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-2\right)^{2}.

$$x^{2}-2x+14+y^{2}-4y+z^{2}-6z=x^{2}-4x+4+\left(y+1\right)^{2}+\left(z-4\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(y+1\right)^{2}.

$$x^{2}-2x+14+y^{2}-4y+z^{2}-6z=x^{2}-4x+4+y^{2}+2y+1+\left(z-4\right)^{2}$$

Add 4 and 1 to get 5.

$$x^{2}-2x+14+y^{2}-4y+z^{2}-6z=x^{2}-4x+5+y^{2}+2y+\left(z-4\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(z-4\right)^{2}.

$$x^{2}-2x+14+y^{2}-4y+z^{2}-6z=x^{2}-4x+5+y^{2}+2y+z^{2}-8z+16$$

Add 5 and 16 to get 21.

$$x^{2}-2x+14+y^{2}-4y+z^{2}-6z=x^{2}-4x+21+y^{2}+2y+z^{2}-8z$$

Subtract x^{2} from both sides.

$$x^{2}-2x+14+y^{2}-4y+z^{2}-6z-x^{2}=-4x+21+y^{2}+2y+z^{2}-8z$$

Combine x^{2} and -x^{2} to get 0.

$$-2x+14+y^{2}-4y+z^{2}-6z=-4x+21+y^{2}+2y+z^{2}-8z$$

Add 4x to both sides.

$$-2x+14+y^{2}-4y+z^{2}-6z+4x=21+y^{2}+2y+z^{2}-8z$$

Combine -2x and 4x to get 2x.

$$2x+14+y^{2}-4y+z^{2}-6z=21+y^{2}+2y+z^{2}-8z$$

Subtract 14 from both sides.

$$2x+y^{2}-4y+z^{2}-6z=21+y^{2}+2y+z^{2}-8z-14$$

Subtract 14 from 21 to get 7.

$$2x+y^{2}-4y+z^{2}-6z=7+y^{2}+2y+z^{2}-8z$$

Subtract y^{2} from both sides.

$$2x-4y+z^{2}-6z=7+y^{2}+2y+z^{2}-8z-y^{2}$$

Combine y^{2} and -y^{2} to get 0.

$$2x-4y+z^{2}-6z=7+2y+z^{2}-8z$$

Add 4y to both sides.

$$2x+z^{2}-6z=7+2y+z^{2}-8z+4y$$

Combine 2y and 4y to get 6y.

$$2x+z^{2}-6z=7+6y+z^{2}-8z$$

Subtract z^{2} from both sides.

$$2x-6z=7+6y+z^{2}-8z-z^{2}$$

Combine z^{2} and -z^{2} to get 0.

$$2x-6z=7+6y-8z$$

Add 6z to both sides.

$$2x=7+6y-8z+6z$$

Combine -8z and 6z to get -2z.

$$2x=7+6y-2z$$

The equation is in standard form.

$$2x=6y-2z+7$$

Divide both sides by 2.

$$\frac{2x}{2}=\frac{6y-2z+7}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x=\frac{6y-2z+7}{2}$$

Divide 7+6y-2z by 2.

$$x=3y-z+\frac{7}{2}$$

Solve for y

$$y=\frac{x}{3}+\frac{z}{3}-\frac{7}{6}$$

Steps for Solving Linear Equation

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

$$x^{2}-2x+1+\left(y-2\right)^{2}+\left(z-3\right)^{2}=\left(x-2\right)^{2}+\left(y+1\right)^{2}+\left(z-4\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(y-2\right)^{2}.

$$x^{2}-2x+1+y^{2}-4y+4+\left(z-3\right)^{2}=\left(x-2\right)^{2}+\left(y+1\right)^{2}+\left(z-4\right)^{2}$$

Add 1 and 4 to get 5.

$$x^{2}-2x+5+y^{2}-4y+\left(z-3\right)^{2}=\left(x-2\right)^{2}+\left(y+1\right)^{2}+\left(z-4\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(z-3\right)^{2}.

$$x^{2}-2x+5+y^{2}-4y+z^{2}-6z+9=\left(x-2\right)^{2}+\left(y+1\right)^{2}+\left(z-4\right)^{2}$$

Add 5 and 9 to get 14.

$$x^{2}-2x+14+y^{2}-4y+z^{2}-6z=\left(x-2\right)^{2}+\left(y+1\right)^{2}+\left(z-4\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-2\right)^{2}.

$$x^{2}-2x+14+y^{2}-4y+z^{2}-6z=x^{2}-4x+4+\left(y+1\right)^{2}+\left(z-4\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(y+1\right)^{2}.

$$x^{2}-2x+14+y^{2}-4y+z^{2}-6z=x^{2}-4x+4+y^{2}+2y+1+\left(z-4\right)^{2}$$

Add 4 and 1 to get 5.

$$x^{2}-2x+14+y^{2}-4y+z^{2}-6z=x^{2}-4x+5+y^{2}+2y+\left(z-4\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(z-4\right)^{2}.

$$x^{2}-2x+14+y^{2}-4y+z^{2}-6z=x^{2}-4x+5+y^{2}+2y+z^{2}-8z+16$$

Add 5 and 16 to get 21.

$$x^{2}-2x+14+y^{2}-4y+z^{2}-6z=x^{2}-4x+21+y^{2}+2y+z^{2}-8z$$

Subtract y^{2} from both sides.

$$x^{2}-2x+14+y^{2}-4y+z^{2}-6z-y^{2}=x^{2}-4x+21+2y+z^{2}-8z$$

Combine y^{2} and -y^{2} to get 0.

$$x^{2}-2x+14-4y+z^{2}-6z=x^{2}-4x+21+2y+z^{2}-8z$$

Subtract 2y from both sides.

$$x^{2}-2x+14-4y+z^{2}-6z-2y=x^{2}-4x+21+z^{2}-8z$$

Combine -4y and -2y to get -6y.

$$x^{2}-2x+14-6y+z^{2}-6z=x^{2}-4x+21+z^{2}-8z$$

Subtract x^{2} from both sides.

$$-2x+14-6y+z^{2}-6z=x^{2}-4x+21+z^{2}-8z-x^{2}$$

Combine x^{2} and -x^{2} to get 0.

$$-2x+14-6y+z^{2}-6z=-4x+21+z^{2}-8z$$

Add 2x to both sides.

$$14-6y+z^{2}-6z=-4x+21+z^{2}-8z+2x$$

Combine -4x and 2x to get -2x.

$$14-6y+z^{2}-6z=-2x+21+z^{2}-8z$$

Subtract 14 from both sides.

$$-6y+z^{2}-6z=-2x+21+z^{2}-8z-14$$

Subtract 14 from 21 to get 7.

$$-6y+z^{2}-6z=-2x+7+z^{2}-8z$$

Subtract z^{2} from both sides.

$$-6y-6z=-2x+7+z^{2}-8z-z^{2}$$

Combine z^{2} and -z^{2} to get 0.

$$-6y-6z=-2x+7-8z$$

Add 6z to both sides.

$$-6y=-2x+7-8z+6z$$

Combine -8z and 6z to get -2z.

$$-6y=-2x+7-2z$$

The equation is in standard form.

$$-6y=7-2z-2x$$

Divide both sides by -6.

$$\frac{-6y}{-6}=\frac{7-2z-2x}{-6}$$

Dividing by -6 undoes the multiplication by -6.

$$y=\frac{7-2z-2x}{-6}$$

Divide -2x+7-2z by -6.

$$y=\frac{x}{3}+\frac{z}{3}-\frac{7}{6}$$