Question

Solve the proportion. Check for extraneous solutions. $$\frac{5}{2 y}=\frac{7}{y-3}$$

Short answer

The solution to the proportion is \(y = - \frac{5}{3}\)

Step-by-step solution

- Cross Multiply

Start by setting the two fractions equal to each other and cross-multiply: \(5*(y-3) = 7*(2y)\).

- Distribute and Simplify

After cross-multiplying, the equation will look like \(5y -15 = 14y\). Now subtract 5y from both sides to isolate y in one side: \(-15 = 9y\).

- Solving for y

The equation now is \(-15 = 9y\). Now divide throughout by 9 to get the value of \(y = -\frac{15}{9} = - \frac{5}{3}\).

- Check for Extraneous Solutions

Now, we plug the found solution \(y = - \frac{5}{3}\) into the original equation and verify if both the left and right sides of the equation match up. The equations after substitution are \(\frac{5}{2(-\frac{5}{3})} = \frac{5}{\frac{-10}{3}} = -\frac{3}{2}\) and \(\frac{7}{-\frac{5}{3}-3} = \frac{7}{-\frac{14}{3}} = \frac{-3}{2}\). As both sides match, there is no extraneous solution. The solution \(y = - \frac{5}{3}\) is valid.