Solve left(-3-xright)^2+left(4-yright)^2=left(6-xright)^2+left(2-yright)^2

$${ \left(-3-x \right) }^{ 2 } + { \left(4-y \right) }^{ 2 } = { \left(6-x \right) }^{ 2 } + { \left(2-y \right) }^{ 2 }$$

Solve for x

$$x=\frac{2y}{9}+\frac{5}{6}$$

Steps for Solving Linear Equation

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

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

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

$$9+6x+x^{2}+16-8y+y^{2}=\left(6-x\right)^{2}+\left(2-y\right)^{2}$$

Add 9 and 16 to get 25.

$$25+6x+x^{2}-8y+y^{2}=\left(6-x\right)^{2}+\left(2-y\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(6-x\right)^{2}.

$$25+6x+x^{2}-8y+y^{2}=36-12x+x^{2}+\left(2-y\right)^{2}$$

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

$$25+6x+x^{2}-8y+y^{2}=36-12x+x^{2}+4-4y+y^{2}$$

Add 36 and 4 to get 40.

$$25+6x+x^{2}-8y+y^{2}=40-12x+x^{2}-4y+y^{2}$$

Add 12x to both sides.

$$25+6x+x^{2}-8y+y^{2}+12x=40+x^{2}-4y+y^{2}$$

Combine 6x and 12x to get 18x.

$$25+18x+x^{2}-8y+y^{2}=40+x^{2}-4y+y^{2}$$

Subtract x^{2} from both sides.

$$25+18x+x^{2}-8y+y^{2}-x^{2}=40-4y+y^{2}$$

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

$$25+18x-8y+y^{2}=40-4y+y^{2}$$

Subtract 25 from both sides.

$$18x-8y+y^{2}=40-4y+y^{2}-25$$

Subtract 25 from 40 to get 15.

$$18x-8y+y^{2}=15-4y+y^{2}$$

Add 8y to both sides.

$$18x+y^{2}=15-4y+y^{2}+8y$$

Combine -4y and 8y to get 4y.

$$18x+y^{2}=15+4y+y^{2}$$

Subtract y^{2} from both sides.

$$18x=15+4y+y^{2}-y^{2}$$

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

$$18x=15+4y$$

The equation is in standard form.

$$18x=4y+15$$

Divide both sides by 18.

$$\frac{18x}{18}=\frac{4y+15}{18}$$

Dividing by 18 undoes the multiplication by 18.

$$x=\frac{4y+15}{18}$$

Divide 15+4y by 18.

$$x=\frac{2y}{9}+\frac{5}{6}$$

Solve for y

$$y=\frac{9x}{2}-\frac{15}{4}$$

Steps for Solving Linear Equation

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

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

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

$$9+6x+x^{2}+16-8y+y^{2}=\left(6-x\right)^{2}+\left(2-y\right)^{2}$$

Add 9 and 16 to get 25.

$$25+6x+x^{2}-8y+y^{2}=\left(6-x\right)^{2}+\left(2-y\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(6-x\right)^{2}.

$$25+6x+x^{2}-8y+y^{2}=36-12x+x^{2}+\left(2-y\right)^{2}$$

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

$$25+6x+x^{2}-8y+y^{2}=36-12x+x^{2}+4-4y+y^{2}$$

Add 36 and 4 to get 40.

$$25+6x+x^{2}-8y+y^{2}=40-12x+x^{2}-4y+y^{2}$$

Add 4y to both sides.

$$25+6x+x^{2}-8y+y^{2}+4y=40-12x+x^{2}+y^{2}$$

Combine -8y and 4y to get -4y.

$$25+6x+x^{2}-4y+y^{2}=40-12x+x^{2}+y^{2}$$

Subtract y^{2} from both sides.

$$25+6x+x^{2}-4y+y^{2}-y^{2}=40-12x+x^{2}$$

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

$$25+6x+x^{2}-4y=40-12x+x^{2}$$

Subtract 25 from both sides.

$$6x+x^{2}-4y=40-12x+x^{2}-25$$

Subtract 25 from 40 to get 15.

$$6x+x^{2}-4y=15-12x+x^{2}$$

Subtract 6x from both sides.

$$x^{2}-4y=15-12x+x^{2}-6x$$

Combine -12x and -6x to get -18x.

$$x^{2}-4y=15-18x+x^{2}$$

Subtract x^{2} from both sides.

$$-4y=15-18x+x^{2}-x^{2}$$

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

$$-4y=15-18x$$

Divide both sides by -4.

$$\frac{-4y}{-4}=\frac{15-18x}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$y=\frac{15-18x}{-4}$$

Divide 15-18x by -4.

$$y=\frac{9x}{2}-\frac{15}{4}$$