- Identify the domain restriction: \(v
eq 4\).
- Simplify the equation by multiplying both sides by the common denominator \(3(v-4)\).
- Solve the resulting linear equation \(-12v + 46 = -21\) for \(v\).
- The final solution is \(v = \boxed{\frac{67}{12}}\).
Explanation
- Analyzing the Equation and Domain
We are given the equation:
\[-\frac{2}{3v - 12} - 4 = -\frac{7}{v - 4}.\]
First, let's analyze the expression in the denominators. Notice that \(3v - 12 = 3(v - 4)\). The equation can be rewritten as:
\[-\frac{2}{3(v - 4)} - 4 = -\frac{7}{v - 4}.\]
Before solving, we must identify the values for which the expression is undefined. The denominators are zero when \(v - 4 = 0\), which means \(v = 4\). Therefore, we must have the restriction \(v
eq 4\).
- Clearing the Denominators
To eliminate the denominators, we multiply the entire equation by the least common denominator, which is \(3(v - 4)\):
\[3(v - 4) \times \left(-\frac{2}{3(v - 4)} - 4\right) = 3(v - 4) \times \left(-\frac{7}{v - 4}\right).\]
Distributing the \(3(v - 4)\) on both sides:
\[3(v - 4) \left(-\frac{2}{3(v - 4)}\right) - 3(v - 4)(4) = 3(v - 4) \left(-\frac{7}{v - 4}\right).\]
Simplifying the terms:
\[-2 - 12(v - 4) = -21.\]
- Solving for v
Now, we simplify the left side of the equation:
\[-2 - 12v + 48 = -21.\]
Combine the constant terms on the left:
\[-12v + 46 = -21.\]
Next, isolate the term involving \(v\) by subtracting \(46\) from both sides:
\[-12v = -21 - 46\]
\[-12v = -67.\]
Finally, solve for \(v\) by dividing by \(-12\):
\[v = \frac{-67}{-12} = \frac{67}{12}.\]
- Verification and Conclusion
We have found \(v = \frac{67}{12}\). We must check if this value satisfies the original domain restriction \(v
eq 4\). Since \(\frac{67}{12} \approx 5.58\), which is not equal to \(4\), the solution is valid. Thus, the value for \(v\) is \(\frac{67}{12}\).
Examples
Equations like this often appear in physics or engineering problems where you need to solve for an unknown quantity like time or distance while working with rational expressions. For instance, if you are modeling the time taken for two tasks to be completed relative to a certain offset, you might encounter a formula where denominators represent rates or distances. Solving this type of equation allows you to find the exact point where these rates balance out.