Solve {l}{sqrt{2014*2016+1}=x}{sqrt{x(x+4)+4}=y}{sqrt{(y+2)*(y+8)+9}=z}

$$\left. \begin{array}{l}{ \sqrt { 2014 \cdot 2016 + 1 } = x }\\{ \sqrt { x ( x + 4 ) + 4 } = y }\\{ \sqrt { ( y + 2 ) \cdot ( y + 8 ) + 9 } = z }\end{array} \right.$$

Solve for x, y, z

$$x=2015$$

$$y=2017$$

$$z=2022$$

Solution Steps

Consider the first equation. Multiply 2014 and 2016 to get 4060224.

$$\sqrt{4060224+1}=x$$

Add 4060224 and 1 to get 4060225.

$$\sqrt{4060225}=x$$

Calculate the square root of 4060225 and get 2015.

$$2015=x$$

Swap sides so that all variable terms are on the left hand side.

$$x=2015$$

Consider the second equation. Insert the known values of variables into the equation.

$$\sqrt{2015\left(2015+4\right)+4}=y$$

Add 2015 and 4 to get 2019.

$$\sqrt{2015\times 2019+4}=y$$

Multiply 2015 and 2019 to get 4068285.

$$\sqrt{4068285+4}=y$$

Add 4068285 and 4 to get 4068289.

$$\sqrt{4068289}=y$$

Calculate the square root of 4068289 and get 2017.

$$2017=y$$

Swap sides so that all variable terms are on the left hand side.

$$y=2017$$

Consider the third equation. Insert the known values of variables into the equation.

$$\sqrt{\left(2017+2\right)\left(2017+8\right)+9}=z$$

Add 2017 and 2 to get 2019.

$$\sqrt{2019\left(2017+8\right)+9}=z$$

Add 2017 and 8 to get 2025.

$$\sqrt{2019\times 2025+9}=z$$

Multiply 2019 and 2025 to get 4088475.

$$\sqrt{4088475+9}=z$$

Add 4088475 and 9 to get 4088484.

$$\sqrt{4088484}=z$$

Calculate the square root of 4088484 and get 2022.

$$2022=z$$

Swap sides so that all variable terms are on the left hand side.

$$z=2022$$

The system is now solved.

$$x=2015 y=2017 z=2022$$