Solve {c}{C_{1}+C_{2}=1}{frac{1+sqrt{5}}{2}C_{1}+frac{1-sqrt{5}}{2}C_{2}=2}

$$\left. \begin{array} { c } { C _ { 1 } + C _ { 2 } = 1 } \\ { \frac { 1 + \sqrt { 5 } } { 2 } C _ { 1 } + \frac { 1 - \sqrt { 5 } } { 2 } C _ { 2 } = 2 } \end{array} \right.$$

Solve for C_1, C_2

$$C_{1}=\frac{3\sqrt{5}}{10}+\frac{1}{2}\approx 1.170820393$$

$$C_{2}=-\frac{3\sqrt{5}}{10}+\frac{1}{2}\approx -0.170820393$$

Steps Using Substitution

To solve a pair of equations using substitution, first solve one of the equations for one of the variables. Then substitute the result for that variable in the other equation.

$$C_{1}+C_{2}=1,\frac{\sqrt{5}+1}{2}C_{1}+\frac{1-\sqrt{5}}{2}C_{2}=2$$

Choose one of the equations and solve it for C_{1} by isolating C_{1} on the left hand side of the equal sign.

$$C_{1}+C_{2}=1$$

Subtract C_{2} from both sides of the equation.

$$C_{1}=-C_{2}+1$$

Substitute -C_{2}+1 for C_{1} in the other equation, \frac{\sqrt{5}+1}{2}C_{1}+\frac{1-\sqrt{5}}{2}C_{2}=2.

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

Multiply \frac{1}{2}+\frac{1}{2}\sqrt{5} times -C_{2}+1.

$$\frac{-\sqrt{5}-1}{2}C_{2}+\frac{\sqrt{5}+1}{2}+\frac{1-\sqrt{5}}{2}C_{2}=2$$

Add \frac{-C_{2}-C_{2}\sqrt{5}}{2} to \frac{\left(1-\sqrt{5}\right)C_{2}}{2}.

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

Subtract \frac{1+\sqrt{5}}{2} from both sides of the equation.

$$\left(-\sqrt{5}\right)C_{2}=\frac{3-\sqrt{5}}{2}$$

Divide both sides by -\sqrt{5}.

$$C_{2}=-\frac{3\sqrt{5}}{10}+\frac{1}{2}$$

Substitute -\frac{3\sqrt{5}}{10}+\frac{1}{2} for C_{2} in C_{1}=-C_{2}+1. Because the resulting equation contains only one variable, you can solve for C_{1} directly.

$$C_{1}=-\left(-\frac{3\sqrt{5}}{10}+\frac{1}{2}\right)+1$$

Multiply -1 times -\frac{3\sqrt{5}}{10}+\frac{1}{2}.

$$C_{1}=\frac{3\sqrt{5}}{10}-\frac{1}{2}+1$$

Add 1 to \frac{3\sqrt{5}}{10}-\frac{1}{2}.

$$C_{1}=\frac{3\sqrt{5}}{10}+\frac{1}{2}$$

The system is now solved.

$$C_{1}=\frac{3\sqrt{5}}{10}+\frac{1}{2},C_{2}=-\frac{3\sqrt{5}}{10}+\frac{1}{2}$$

Steps Using Elimination

In order to solve by elimination, coefficients of one of the variables must be the same in both equations so that the variable will cancel out when one equation is subtracted from the other.

$$C_{1}+C_{2}=1,\frac{\sqrt{5}+1}{2}C_{1}+\frac{1-\sqrt{5}}{2}C_{2}=2$$

To make C_{1} and \frac{\left(1+\sqrt{5}\right)C_{1}}{2} equal, multiply all terms on each side of the first equation by \frac{1}{2}+\frac{1}{2}\sqrt{5} and all terms on each side of the second by 1.

$$\frac{\sqrt{5}+1}{2}C_{1}+\frac{\sqrt{5}+1}{2}C_{2}=\frac{\sqrt{5}+1}{2},\frac{\sqrt{5}+1}{2}C_{1}+\frac{1-\sqrt{5}}{2}C_{2}=2$$

Subtract \frac{\sqrt{5}+1}{2}C_{1}+\frac{1-\sqrt{5}}{2}C_{2}=2 from \frac{\sqrt{5}+1}{2}C_{1}+\frac{\sqrt{5}+1}{2}C_{2}=\frac{\sqrt{5}+1}{2} by subtracting like terms on each side of the equal sign.

$$\frac{\sqrt{5}+1}{2}C_{1}+\frac{-\sqrt{5}-1}{2}C_{1}+\frac{\sqrt{5}+1}{2}C_{2}+\frac{\sqrt{5}-1}{2}C_{2}=\frac{\sqrt{5}+1}{2}-2$$

Add \frac{\left(1+\sqrt{5}\right)C_{1}}{2} to \frac{-C_{1}-C_{1}\sqrt{5}}{2}. Terms \frac{\left(1+\sqrt{5}\right)C_{1}}{2} and \frac{-C_{1}-C_{1}\sqrt{5}}{2} cancel out, leaving an equation with only one variable that can be solved.

$$\frac{\sqrt{5}+1}{2}C_{2}+\frac{\sqrt{5}-1}{2}C_{2}=\frac{\sqrt{5}+1}{2}-2$$

Add \frac{\left(1+\sqrt{5}\right)C_{2}}{2} to \frac{\left(-1+\sqrt{5}\right)C_{2}}{2}.

$$\sqrt{5}C_{2}=\frac{\sqrt{5}+1}{2}-2$$

Add \frac{1+\sqrt{5}}{2} to -2.

$$\sqrt{5}C_{2}=\frac{\sqrt{5}-3}{2}$$

Divide both sides by \sqrt{5}.

$$C_{2}=-\frac{3\sqrt{5}}{10}+\frac{1}{2}$$

Substitute -\frac{3\sqrt{5}}{10}+\frac{1}{2} for C_{2} in \frac{\sqrt{5}+1}{2}C_{1}+\frac{1-\sqrt{5}}{2}C_{2}=2. Because the resulting equation contains only one variable, you can solve for C_{1} directly.

$$\frac{\sqrt{5}+1}{2}C_{1}+\frac{1-\sqrt{5}}{2}\left(-\frac{3\sqrt{5}}{10}+\frac{1}{2}\right)=2$$

Multiply \frac{1}{2}-\frac{1}{2}\sqrt{5} times -\frac{3\sqrt{5}}{10}+\frac{1}{2}.

$$\frac{\sqrt{5}+1}{2}C_{1}-\frac{2\sqrt{5}}{5}+1=2$$

Subtract -\frac{2\sqrt{5}}{5}+1 from both sides of the equation.

$$\frac{\sqrt{5}+1}{2}C_{1}=\frac{2\sqrt{5}}{5}+1$$

Divide both sides by \frac{1}{2}+\frac{1}{2}\sqrt{5}.

$$C_{1}=\frac{3\sqrt{5}}{10}+\frac{1}{2}$$

The system is now solved.

$$C_{1}=\frac{3\sqrt{5}}{10}+\frac{1}{2},C_{2}=-\frac{3\sqrt{5}}{10}+\frac{1}{2}$$