Verified Answer :

  • Factor the denominator: \(\frac{5}{2(y-6)} - 2 = \frac{7}{y-6}\), with \(y eq 6\).
  • Multiply by the common denominator \(2(y-6)\) to get \(5 - 4(y-6) = 14\).
  • Simplify the linear equation: \(29 - 4y = 14\).
  • Solve for the final value: \(\boxed{y = \frac{15}{4}}\).

Explanation

  1. Analysis of the Equation and Domain Constraints

The equation provided is \(\frac{5}{2y-12} - 2 = \frac{7}{y-6}\). First, analyze the denominators to find the common denominator and domain constraints. The expression \(2y-12\) can be factored as \(2(y-6)\). Therefore, the equation is \(\frac{5}{2(y-6)} - 2 = \frac{7}{y-6}\). Since division by zero is undefined, the denominator \(y-6\) cannot be zero, which implies \(y eq 6\).

  1. Eliminating the Fractions

To eliminate the fractions, multiply both sides of the equation by the least common denominator, which is \(2(y-6)\). Multiplying each term by \(2(y-6)\) gives:
\[2(y-6) \cdot \left(\frac{5}{2(y-6)}\right) - 2(y-6) \cdot 2 = 2(y-6) \cdot \left(\frac{7}{y-6}\right)\]
This simplifies to:
\[5 - 4(y-6) = 14\]

  1. Solving the Resulting Linear Equation

Now, distribute the \(-4\) into the parenthesis:
\[5 - 4y + 24 = 14\]
Combine the constant terms:
\[29 - 4y = 14\]
Subtract 29 from both sides:
\[-4y = 14 - 29\]
\[-4y = -15\]
Finally, divide by \(-4\):
\[y = \frac{-15}{-4} = \frac{15}{4} = 3.75\]

  1. Conclusion and Verification

The value obtained is \(y = 3.75\). Since \(3.75 eq 6\), this value is a valid solution. Thus, the solution to the equation is \(y = \frac{15}{4}\).

Examples

This type of equation is common in problems involving work rates or mixing proportions where the total time or rate is dependent on variable inputs. For example, if two pumps fill a tank at different rates related to the same variable \(y\), solving an equation like this helps find the exact value of \(y\) needed to reach a specific equilibrium. This method is fundamental for optimizing resource allocation in engineering and logistics.