Solve and check the following equation. \[ \frac{x}{3}+\frac{x}{5}=8 \] The solution set is (Type an integer or a simplified fraction.)

Step 1 ：\(\frac{5x}{15} + \frac{3x}{15} = 8\)

Step 2 ：\(\frac{5x+3x}{15} = 8\)

Step 3 ：\(\frac{8x}{15} = 8\)

Step 4 ：\(8x = 120\)

Step 5 ：\(x = 15\)

Step 6 ：\(\frac{15}{3} + \frac{15}{5} = 8\)

Step 7 ：\(5 + 3 = 8\)

Step 8 ：\(\boxed{15}\)