How were these two equations rearranged to this from?

Question

$$f(x)=F(x)+G(x)$$ $$g(x)=-cF(x)+cG(x)$$

To:

$$F(x)= \frac{1}{2}f(x)-\frac{1}{2c}g(x)$$

$$G(x)= \frac{1}{2}f(x)+\frac{1}{2c}g(x)$$

I can kind of see the relationship between the equations but if someone could explain to me where does the $\frac{1}{2}$ come from I would be thankful.

Answer 1

Dividing the second equation by $c$ yields

$$f(x) = F(x) + G(x)$$ $$\frac{1}{c}g(x) = -F(x) + G(x)$$

Subtracting the second equation from the first yields

$$f(x) - \frac{1}{c} g(x) = 2F(x)$$ $$F(x) = \frac{1}{2}f(x) - \frac{1}{2c} g(x)$$

Adding the second equation to the first yields

$$f(x) + \frac{1}{c} g(x) = 2G(x)$$ $$G(x) = \frac{1}{2}f(x) + \frac{1}{2c} g(x)$$