- Expand both sides of the equation: \(-3p - 15 = 2p + 5\).
- Move terms to isolate \(p\): \(-20 = 5p\).
- Solve for \(p\): \(p = -4\).
- The correct statement is: \(\boxed{\text{The only solution is } p = -4}\).
Explanation
- Analysis of the Equation
The given equation is \(-3(p+5) = 2(p+2) + 1\). We need to determine the value of \(p\) that satisfies this equation, or determine the nature of its solution set. This is a linear equation in one variable, \(p\). To find the solution, we will expand the expressions on both sides, simplify, and then isolate the variable \(p\).
- Expansion and Simplification
First, we expand the terms on both sides of the equation:
Left side: \(-3(p + 5) = -3p - 15\)
Right side: \(2(p + 2) + 1 = 2p + 4 + 1 = 2p + 5\)
Equating the expanded forms, we have:
\(-3p - 15 = 2p + 5\)
- Isolating the Variable p
To solve for \(p\), we gather all terms containing \(p\) on one side and the constant terms on the other:
Add \(3p\) to both sides: \(-15 = 5p + 5\)
Subtract 5 from both sides: \(-15 - 5 = 5p\)
\(-20 = 5p\)
Divide by 5: \(p = \frac{-20}{5} = -4\)
- Verification and Conclusion
We have found that the equation simplifies to \(p = -4\). Substituting \(p = -4\) back into the original equation to verify:
Left side: \(-3(-4 + 5) = -3(1) = -3\)
Right side: \(2(-4 + 2) + 1 = 2(-2) + 1 = -4 + 1 = -3\)
Since both sides are equal, the unique solution to the equation is \(p = -4\). Comparing this to the given options, the correct statement is that the only solution is \(p = -4\).
Examples
In daily life, solving for a single variable is useful for finding unknown quantities in budget planning or resource allocation. For example, if you are determining how many hours \(p\) you need to work to cover a cost, where the costs follow a linear relationship such as \(-3(p + 5) = 2(p + 2) + 1\), finding the correct value of \(p\) helps you manage your time and finances effectively. By isolating the variable, you can make informed decisions based on calculated data.