Questions: Solve the following proportion. 6x/(x+5) = 5x/(x+4)

Solution Steps

Cross-multiply the given proportion: \[ \frac{6x}{x+5} = \frac{5x}{x+4} \] \[ 6x(x+4) = 5x(x+5) \]

Expand both sides of the equation: \[ 6x^2 + 24x = 5x^2 + 25x \]

Subtract \(5x^2\) and \(24x\) from both sides: \[ 6x^2 - 5x^2 + 24x - 24x = 5x^2 - 5x^2 + 25x - 24x \] \[ x^2 = x \] \[ x^2 - x = 0 \]

Factor the quadratic equation: \[ x(x-1) = 0 \]

Set each factor equal to zero and solve for \(x\): \[ x = 0 \quad \text{or} \quad x-1 = 0 \] \[ x = 0 \quad \text{or} \quad x = 1 \]

Check if any of the solutions make the denominator zero in the original equation. If \(x=0\), then \(x+5 = 5\) and \(x+4 = 4\), which are both non-zero. If \(x=1\), then \(x+5 = 6\) and \(x+4 = 5\), which are both non-zero. Thus, both solutions are valid.

Final Answer