Solve x^2+1-2x+y^2+2y+1+z^2-4z+4=x^2+{y}^2-2y+1+{z}^2-6z+9

$${ x }^{ 2 } +1-2x+ { y }^{ 2 } +2y+1+ { z }^{ 2 } -4z+4= { x }^{ 2 } + { y }^{ 2 } -2y+1+ { z }^{ 2 } -6z+9$$

Solve for x

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

Steps for Solving Linear Equation

Add 1 and 1 to get 2.

$$x^{2}+2-2x+y^{2}+2y+z^{2}-4z+4=x^{2}+y^{2}-2y+1+z^{2}-6z+9$$

Add 2 and 4 to get 6.

$$x^{2}+6-2x+y^{2}+2y+z^{2}-4z=x^{2}+y^{2}-2y+1+z^{2}-6z+9$$

Add 1 and 9 to get 10.

$$x^{2}+6-2x+y^{2}+2y+z^{2}-4z=x^{2}+y^{2}-2y+10+z^{2}-6z$$

Subtract x^{2} from both sides.

$$x^{2}+6-2x+y^{2}+2y+z^{2}-4z-x^{2}=y^{2}-2y+10+z^{2}-6z$$

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

$$6-2x+y^{2}+2y+z^{2}-4z=y^{2}-2y+10+z^{2}-6z$$

Subtract 6 from both sides.

$$-2x+y^{2}+2y+z^{2}-4z=y^{2}-2y+10+z^{2}-6z-6$$

Subtract 6 from 10 to get 4.

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

Subtract y^{2} from both sides.

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

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

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

Subtract 2y from both sides.

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

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

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

Subtract z^{2} from both sides.

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

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

$$-2x-4z=-4y+4-6z$$

Add 4z to both sides.

$$-2x=-4y+4-6z+4z$$

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

$$-2x=-4y+4-2z$$

The equation is in standard form.

$$-2x=4-2z-4y$$

Divide both sides by -2.

$$\frac{-2x}{-2}=\frac{4-2z-4y}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x=\frac{4-2z-4y}{-2}$$

Divide -4y+4-2z by -2.

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

Solve for y

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

Steps for Solving Linear Equation

Add 1 and 1 to get 2.

$$x^{2}+2-2x+y^{2}+2y+z^{2}-4z+4=x^{2}+y^{2}-2y+1+z^{2}-6z+9$$

Add 2 and 4 to get 6.

$$x^{2}+6-2x+y^{2}+2y+z^{2}-4z=x^{2}+y^{2}-2y+1+z^{2}-6z+9$$

Add 1 and 9 to get 10.

$$x^{2}+6-2x+y^{2}+2y+z^{2}-4z=x^{2}+y^{2}-2y+10+z^{2}-6z$$

Subtract y^{2} from both sides.

$$x^{2}+6-2x+y^{2}+2y+z^{2}-4z-y^{2}=x^{2}-2y+10+z^{2}-6z$$

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

$$x^{2}+6-2x+2y+z^{2}-4z=x^{2}-2y+10+z^{2}-6z$$

Add 2y to both sides.

$$x^{2}+6-2x+2y+z^{2}-4z+2y=x^{2}+10+z^{2}-6z$$

Combine 2y and 2y to get 4y.

$$x^{2}+6-2x+4y+z^{2}-4z=x^{2}+10+z^{2}-6z$$

Subtract x^{2} from both sides.

$$6-2x+4y+z^{2}-4z=x^{2}+10+z^{2}-6z-x^{2}$$

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

$$6-2x+4y+z^{2}-4z=10+z^{2}-6z$$

Subtract 6 from both sides.

$$-2x+4y+z^{2}-4z=10+z^{2}-6z-6$$

Subtract 6 from 10 to get 4.

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

Add 2x to both sides.

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

Subtract z^{2} from both sides.

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

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

$$4y-4z=4-6z+2x$$

Add 4z to both sides.

$$4y=4-6z+2x+4z$$

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

$$4y=4-2z+2x$$

The equation is in standard form.

$$4y=2x-2z+4$$

Divide both sides by 4.

$$\frac{4y}{4}=\frac{2x-2z+4}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$y=\frac{2x-2z+4}{4}$$

Divide 4-2z+2x by 4.

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