4/n=1/a+1/b+1/c Solve the equation

To solve the equation \[ \frac{4}{n} = \frac{1}{a} + \frac{1}{b} + \frac{1}{c}, \] let's first get a common denominator for the right-hand side. The common denominator for \( a \), \( b \), and \( c \) is \( abc \). Thus, we can rewrite the right-hand side: \[ \frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{bc}{abc} + \frac{ac}{abc} + \frac{ab}{abc} = \frac{bc + ac + ab}{abc}. \] Now we can rewrite the original equation as: \[ \frac{4}{n} = \frac{bc + ac + ab}{abc}. \] Next, cross-multiply to eliminate the fractions: \[ 4 \cdot abc = n(bc + ac + ab). \] This gives us the equation: \[ n(bc + ac + ab) = 4abc. \] Now, we can solve for \( n \): \[ n = \frac{4abc}{bc + ac + ab}. \] This expression provides a relationship for \( n \) in terms of \( a \), \( b \), and \( c \). The solution \( n \) depends on the values of \( a \), \( b \), and \( c \).