Question

Solve the proportion. Check for extraneous solutions. $$\frac{6 x-7}{4}=\frac{5}{x}$$

Short answer

The solution to the equation is \( x = -5/2 \) and \( x = 4/3 \)

Step-by-step solution

Cross-Multiplication

Cross-multiplication involves multiplying the numerator of the first fraction by the denominator of the second fraction and setting it equal to the product of the numerator of the second fraction and the denominator of the first fraction. This yields: \( 4 * 5 = (6x - 7) * x \) which simplifies to \( 20 = 6x^2 - 7x \).

Simplify the equation

Rearranging the equation into the standard quadratic form gives: \( 6x^2 - 7x - 20 = 0 \)

Solve for x

This is a quadratic equation, so solve for x by either factoring, completing the square or the quadratic formula. The factoring solution is given by \( (2x + 5)(3x - 4) = 0 \). Setting each factor equal to zero yields the possible solutions of \( x = -5/2 \) and \( x = 4/3 \).

Check for extraneous solutions

An extraneous solution is a solution that causes the denominator of the original expression to be zero. In this equation, neither \( x = -5/2 \) or \( x = 4/3 \) makes the denominator 0.