Solve {l}{x=6}{y=sqrt{760-4x^2}}{z=x-20}{text{Solvefor}a,btext{where}}{a=y}{b=z}

$$\left. \begin{array} { l } { x = 6 }\\ { y = \sqrt{760 - 4 x ^ {2}} }\\ { z = x - 20 }\\ { \text{Solve for } a,b \text{ where} } \\ { a = y }\\ { b = z } \end{array} \right.$$

Solve for x, y, z, a, b

$$a=2\sqrt{154}\approx 24.819347292$$

$$b=-14$$

Solution Steps

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

$$y=\sqrt{760-4\times 6^{2}}$$

Calculate 6 to the power of 2 and get 36.

$$y=\sqrt{760-4\times 36}$$

Multiply -4 and 36 to get -144.

$$y=\sqrt{760-144}$$

Subtract 144 from 760 to get 616.

$$y=\sqrt{616}$$

Factor 616=2^{2}\times 154. Rewrite the square root of the product \sqrt{2^{2}\times 154} as the product of square roots \sqrt{2^{2}}\sqrt{154}. Take the square root of 2^{2}.

$$y=2\sqrt{154}$$

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

$$z=6-20$$

Subtract 20 from 6 to get -14.

$$z=-14$$

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

$$a=2\sqrt{154}$$

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

$$b=-14$$

The system is now solved.

$$x=6 y=2\sqrt{154} z=-14 a=2\sqrt{154} b=-14$$