Solve (7-x)^2+(1-y)^2=(3-x)^2+(5-y^2)

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

Solve for x

$$x=\frac{y^{2}-y+18}{4}$$

Steps for Solving Linear Equation

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(7-x\right)^{2}.

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

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

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

Add 49 and 1 to get 50.

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

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

$$50-14x+x^{2}-2y+y^{2}=9-6x+x^{2}+5-y^{2}$$

Add 9 and 5 to get 14.

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

Add 6x to both sides.

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

Combine -14x and 6x to get -8x.

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

Subtract x^{2} from both sides.

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

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

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

Subtract 50 from both sides.

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

Subtract 50 from 14 to get -36.

$$-8x-2y+y^{2}=-36-y^{2}$$

Add 2y to both sides.

$$-8x+y^{2}=-36-y^{2}+2y$$

Subtract y^{2} from both sides.

$$-8x=-36-y^{2}+2y-y^{2}$$

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

$$-8x=-36-2y^{2}+2y$$

The equation is in standard form.

$$-8x=-2y^{2}+2y-36$$

Divide both sides by -8.

$$\frac{-8x}{-8}=\frac{-2y^{2}+2y-36}{-8}$$

Dividing by -8 undoes the multiplication by -8.

$$x=\frac{-2y^{2}+2y-36}{-8}$$

Divide -36-2y^{2}+2y by -8.

$$x=\frac{y^{2}}{4}-\frac{y}{4}+\frac{9}{2}$$

Solve for y (complex solution)

$$y=\frac{\sqrt{16x-71}+1}{2}$$

$$y=\frac{-\sqrt{16x-71}+1}{2}$$

Solve for y

$$y=\frac{\sqrt{16x-71}+1}{2}$$

$$y=\frac{-\sqrt{16x-71}+1}{2}\text{, }x\geq \frac{71}{16}$$