Solve {l}{b=frac{1+sqrt{5}}{2}a}{a=21}{c=frac{1+sqrt{5}}{2}b}

$$\left\{ \begin{array} { l } { b = \frac { 1 + \sqrt { 5 } } { 2 } a } \\ { a = 21 } \\ { c = \frac { 1 + \sqrt { 5 } } { 2 } b } \end{array} \right.$$

Solve for b, a, c

$$b = \frac{21 {(\sqrt{5} + 1)}}{2} \approx 33.978713764$$

$$a=21$$

$$c = \frac{21 {(\sqrt{5} + 3)}}{2} \approx 54.978713764$$

Short Solution Steps

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

$$b=\frac{1+\sqrt{5}}{2}\times 21$$

Express \frac{1+\sqrt{5}}{2}\times 21 as a single fraction.

$$b=\frac{\left(1+\sqrt{5}\right)\times 21}{2}$$

Use the distributive property to multiply 1+\sqrt{5} by 21.

$$b=\frac{21+21\sqrt{5}}{2}$$

Divide each term of 21+21\sqrt{5} by 2 to get \frac{21}{2}+\frac{21}{2}\sqrt{5}.

$$b=\frac{21}{2}+\frac{21}{2}\sqrt{5}$$

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

$$c=\frac{1+\sqrt{5}}{2}\left(\frac{21}{2}+\frac{21}{2}\sqrt{5}\right)$$

Use the distributive property to multiply \frac{1+\sqrt{5}}{2} by \frac{21}{2}+\frac{21}{2}\sqrt{5}.

$$c=\frac{21}{2}\times \frac{1+\sqrt{5}}{2}+\frac{21}{2}\times \frac{1+\sqrt{5}}{2}\sqrt{5}$$

Multiply \frac{21}{2} times \frac{1+\sqrt{5}}{2} by multiplying numerator times numerator and denominator times denominator.

$$c=\frac{21\left(1+\sqrt{5}\right)}{2\times 2}+\frac{21}{2}\times \frac{1+\sqrt{5}}{2}\sqrt{5}$$

Multiply \frac{21}{2} times \frac{1+\sqrt{5}}{2} by multiplying numerator times numerator and denominator times denominator.

$$c=\frac{21\left(1+\sqrt{5}\right)}{2\times 2}+\frac{21\left(1+\sqrt{5}\right)}{2\times 2}\sqrt{5}$$

Express \frac{21\left(1+\sqrt{5}\right)}{2\times 2}\sqrt{5} as a single fraction.

$$c=\frac{21\left(1+\sqrt{5}\right)}{2\times 2}+\frac{21\left(1+\sqrt{5}\right)\sqrt{5}}{2\times 2}$$

Since \frac{21\left(1+\sqrt{5}\right)}{2\times 2} and \frac{21\left(1+\sqrt{5}\right)\sqrt{5}}{2\times 2} have the same denominator, add them by adding their numerators.

$$c=\frac{21\left(1+\sqrt{5}\right)+21\left(1+\sqrt{5}\right)\sqrt{5}}{2\times 2}$$

Do the multiplications in 21\left(1+\sqrt{5}\right)+21\left(1+\sqrt{5}\right)\sqrt{5}.

$$c=\frac{21+21\sqrt{5}+21\sqrt{5}+105}{2\times 2}$$

Do the calculations in 21+21\sqrt{5}+21\sqrt{5}+105.

$$c=\frac{126+42\sqrt{5}}{2\times 2}$$

Multiply 2 and 2 to get 4.

$$c=\frac{126+42\sqrt{5}}{4}$$

Divide each term of 126+42\sqrt{5} by 4 to get \frac{63}{2}+\frac{21}{2}\sqrt{5}.

$$c=\frac{63}{2}+\frac{21}{2}\sqrt{5}$$

The system is now solved.

$$b=\frac{21}{2}+\frac{21}{2}\sqrt{5} a=21 c=\frac{63}{2}+\frac{21}{2}\sqrt{5}$$