$$-\frac{1}{2}c\left(3^{\frac{1}{2}}-5^{\frac{1}{2}}\right)\times 2=a\sqrt{3}-b\sqrt{5}$$

$$\left(-\frac{1}{2}c\times 3^{\frac{1}{2}}+\frac{1}{2}\times 5^{\frac{1}{2}}c\right)\times 2=a\sqrt{3}-b\sqrt{5}$$

$$-c\times 3^{\frac{1}{2}}+c\times 5^{\frac{1}{2}}=a\sqrt{3}-b\sqrt{5}$$

$$a\sqrt{3}-b\sqrt{5}=-c\times 3^{\frac{1}{2}}+c\times 5^{\frac{1}{2}}$$

$$a\sqrt{3}=-c\times 3^{\frac{1}{2}}+c\times 5^{\frac{1}{2}}+b\sqrt{5}$$

$$\sqrt{3}a=\sqrt{5}b+\sqrt{5}c-\sqrt{3}c$$

$$\frac{\sqrt{3}a}{\sqrt{3}}=\frac{\sqrt{5}b+\sqrt{5}c-\sqrt{3}c}{\sqrt{3}}$$

$$a=\frac{\sqrt{5}b+\sqrt{5}c-\sqrt{3}c}{\sqrt{3}}$$

$$a=\frac{\sqrt{3}\left(\sqrt{5}b+\sqrt{5}c-\sqrt{3}c\right)}{3}$$

$$-\frac{1}{2}c\left(3^{\frac{1}{2}}-5^{\frac{1}{2}}\right)\times 2=a\sqrt{3}-b\sqrt{5}$$

$$\left(-\frac{1}{2}c\times 3^{\frac{1}{2}}+\frac{1}{2}\times 5^{\frac{1}{2}}c\right)\times 2=a\sqrt{3}-b\sqrt{5}$$

$$-c\times 3^{\frac{1}{2}}+c\times 5^{\frac{1}{2}}=a\sqrt{3}-b\sqrt{5}$$

$$a\sqrt{3}-b\sqrt{5}=-c\times 3^{\frac{1}{2}}+c\times 5^{\frac{1}{2}}$$

$$\sqrt{3}a-\sqrt{5}b=\sqrt{5}c-\sqrt{3}c$$

$$-\sqrt{5}b=\sqrt{5}c-\sqrt{3}c-\sqrt{3}a$$

$$\left(-\sqrt{5}\right)b=\sqrt{5}c-\sqrt{3}a-\sqrt{3}c$$

$$\frac{\left(-\sqrt{5}\right)b}{-\sqrt{5}}=\frac{\sqrt{5}c-\sqrt{3}a-\sqrt{3}c}{-\sqrt{5}}$$

$$b=\frac{\sqrt{5}c-\sqrt{3}a-\sqrt{3}c}{-\sqrt{5}}$$

$$b=\frac{\sqrt{15}a}{5}+\frac{\sqrt{15}c}{5}-c$$