Solve sqrt{(2-1)^2+(y-2)^2}=y

$$\sqrt { ( 2 - 1 ) ^ { 2 } + ( y - 2 ) ^ { 2 } } = y$$

Solve for y

$$y = \frac{5}{4} = 1\frac{1}{4} = 1.25$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{\left(2-1\right)^{2}+\left(y-2\right)^{2}}\right)^{2}=y^{2}$$

Subtract 1 from 2 to get 1.

$$\left(\sqrt{1^{2}+\left(y-2\right)^{2}}\right)^{2}=y^{2}$$

Calculate 1 to the power of 2 and get 1.

$$\left(\sqrt{1+\left(y-2\right)^{2}}\right)^{2}=y^{2}$$

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

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

Add 1 and 4 to get 5.

$$\left(\sqrt{5+y^{2}-4y}\right)^{2}=y^{2}$$

Calculate \sqrt{5+y^{2}-4y} to the power of 2 and get 5+y^{2}-4y.

$$5+y^{2}-4y=y^{2}$$

Subtract y^{2} from both sides.

$$5+y^{2}-4y-y^{2}=0$$

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

$$5-4y=0$$

Subtract 5 from both sides. Anything subtracted from zero gives its negation.

$$-4y=-5$$

Divide both sides by -4.

$$y=\frac{-5}{-4}$$

Fraction \frac{-5}{-4} can be simplified to \frac{5}{4} by removing the negative sign from both the numerator and the denominator.

$$y=\frac{5}{4}$$

Substitute \frac{5}{4} for y in the equation \sqrt{\left(2-1\right)^{2}+\left(y-2\right)^{2}}=y.

$$\sqrt{\left(2-1\right)^{2}+\left(\frac{5}{4}-2\right)^{2}}=\frac{5}{4}$$

Simplify. The value y=\frac{5}{4} satisfies the equation.

$$\frac{5}{4}=\frac{5}{4}$$

Equation \sqrt{\left(y-2\right)^{2}+1}=y has a unique solution.

$$y=\frac{5}{4}$$