Questions: Solve the equation -1/x = 5/(3x) - 1 x=

Solution Steps

To solve the rational equation \(\frac{-1}{x} = \frac{5}{3x} - 1\), we need to find a common denominator to combine the fractions on the right side of the equation. Once combined, we can cross-multiply to eliminate the fractions and solve for \(x\).

The given equation is \(\frac{-1}{x} = \frac{5}{3x} - 1\). To combine the fractions on the right side, we need a common denominator. The common denominator for the terms on the right is \(3x\). Rewrite the equation as: \[ \frac{-1}{x} = \frac{5 - 3x}{3x} \]

Cross-multiply to eliminate the fractions: \[ -1 \cdot 3x = x \cdot (5 - 3x) \] This simplifies to: \[ -3x = 5x - 3x^2 \]

Rearrange the terms to form a quadratic equation: \[ 3x^2 - 8x = 0 \]

Factor out the common term \(x\): \[ x(3x - 8) = 0 \]

Set each factor equal to zero and solve for \(x\):

1. \(x = 0\)
2. \(3x - 8 = 0 \Rightarrow 3x = 8 \Rightarrow x = \frac{8}{3}\)

Since \(x = 0\) would make the original equation undefined, we discard it.

Final Answer