Solve {l}{a=34}{b=frac{1+sqrt{5}}{2}a}{c=frac{1+sqrt{5}}{2}b}

$$\left. \begin{array} { l } { a = 34 } \\ { b = \frac { 1 + \sqrt { 5 } } { 2 } a } \\ { c = \frac { 1 + \sqrt { 5 } } { 2 } b } \end{array} \right.$$

Solve for a, b, c

$$a=34$$

$$b=17\left(\sqrt{5}+1\right)\approx 55.013155617$$

$$c=17\left(\sqrt{5}+3\right)\approx 89.013155617$$

Solution Steps

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

$$b=\frac{1+\sqrt{5}}{2}\times 34$$

Cancel out 2, the greatest common factor in 34 and 2.

$$b=17\left(1+\sqrt{5}\right)$$

Use the distributive property to multiply 17 by 1+\sqrt{5}.

$$b=17+17\sqrt{5}$$

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

$$c=\frac{1+\sqrt{5}}{2}\left(17+17\sqrt{5}\right)$$

Express \frac{1+\sqrt{5}}{2}\left(17+17\sqrt{5}\right) as a single fraction.

$$c=\frac{\left(1+\sqrt{5}\right)\left(17+17\sqrt{5}\right)}{2}$$

Use the distributive property to multiply 1+\sqrt{5} by 17+17\sqrt{5} and combine like terms.

$$c=\frac{17+34\sqrt{5}+17\left(\sqrt{5}\right)^{2}}{2}$$

The square of \sqrt{5} is 5.

$$c=\frac{17+34\sqrt{5}+17\times 5}{2}$$

Multiply 17 and 5 to get 85.

$$c=\frac{17+34\sqrt{5}+85}{2}$$

Add 17 and 85 to get 102.

$$c=\frac{102+34\sqrt{5}}{2}$$

Divide each term of 102+34\sqrt{5} by 2 to get 51+17\sqrt{5}.

$$c=51+17\sqrt{5}$$

The system is now solved.

$$a=34 b=17+17\sqrt{5} c=51+17\sqrt{5}$$