- Expand the equation using the distributive property: \(-3v + 21 = 3v + 15 + 6\).
- Simplify the right side: \(-3v + 21 = 3v + 21\).
- Isolate the variable: \(-3v = 3v\), then \(6v = 0\).
- Solve for the variable to find \(v = 0\), identifying that the equation has exactly one solution.
Final answer: \(\boxed{v=0}\)
Explanation
- Expand Both Sides of the Equation
To determine the nature of the solution for the equation \(3(-v + 7) = 3(v + 5) + 6\), we first expand both sides using the distributive property, which states that \(a(b + c) = ab + ac\). On the left side, we multiply 3 by \((-v + 7)\) to get \(3(-v) + 3(7) = -3v + 21\). On the right side, we multiply 3 by \((v + 5)\) to get \(3(v) + 3(5) = 3v + 15\), so the full right side becomes \(3v + 15 + 6\). The equation now stands as \(-3v + 21 = 3v + 15 + 6\).
- Simplify and Solve for v
Next, we simplify the equation by combining the constant terms on the right side. The right side is \(3v + 15 + 6\), which simplifies to \(3v + 21\). Now, our equation is \(-3v + 21 = 3v + 21\). To isolate the variable \(v\), we can subtract 21 from both sides, which gives \(-3v = 3v\). Adding \(3v\) to both sides results in \(0 = 6v\). Dividing both sides by 6, we find \(v = 0\).
- Verification and Conclusion
Having found \(v = 0\), we verify the solution by substituting it back into the original equation:
Left Side: \(3(-(0) + 7) = 3(7) = 21\)
Right Side: \(3(0 + 5) + 6 = 3(5) + 6 = 15 + 6 = 21\)
Since the left side equals the right side, the equation has one solution, which is \(v = 0\).
Examples
In real-life scenarios, such as comparing the costs of two different service providers, you might create an algebraic equation to find the point of equality. If Provider A charges a flat rate and a variable fee based on usage (\(v\)), and Provider B has a different structure, solving for \(v\) helps you determine the exact level of usage where both plans cost the same. This allows you to make an informed decision on which service is more economical for your needs.