Solve sqrt{y^2-9}=9-y

$$\sqrt { y ^ { 2 } - 9 } = 9 - y$$

Solve for y

$$y=5$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{y^{2}-9}\right)^{2}=\left(9-y\right)^{2}$$

Calculate \sqrt{y^{2}-9} to the power of 2 and get y^{2}-9.

$$y^{2}-9=\left(9-y\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(9-y\right)^{2}.

$$y^{2}-9=81-18y+y^{2}$$

Add 18y to both sides.

$$y^{2}-9+18y=81+y^{2}$$

Subtract y^{2} from both sides.

$$y^{2}-9+18y-y^{2}=81$$

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

$$-9+18y=81$$

Add 9 to both sides.

$$18y=81+9$$

Add 81 and 9 to get 90.

$$18y=90$$

Divide both sides by 18.

$$y=\frac{90}{18}$$

Divide 90 by 18 to get 5.

$$y=5$$

Substitute 5 for y in the equation \sqrt{y^{2}-9}=9-y.

$$\sqrt{5^{2}-9}=9-5$$

Simplify. The value y=5 satisfies the equation.

$$4=4$$

Equation \sqrt{y^{2}-9}=9-y has a unique solution.

$$y=5$$