Solve $3 n-5 p+2 n=10 p$ for $n$.
A. $n=3 p$
B. $n=\frac{1}{p}$
C. $n=\frac{p}{3}$
D. $n=p$
Answer (2)
Combine like terms: 3 n + 2 n − 5 p = 5 n − 5 p .
Add 5 p to both sides: 5 n = 15 p .
Divide by 5: n = 3 p .
The solution is n = 3 p .
Explanation
Understanding the Problem We are given the equation 3 n − 5 p + 2 n = 10 p and asked to solve for n in terms of p . This involves algebraic manipulation to isolate n on one side of the equation.
Combining Like Terms First, combine the like terms on the left side of the equation: 3 n + 2 n − 5 p = 10 p . This simplifies to 5 n − 5 p = 10 p .
Isolating the Term with n Next, add 5 p to both sides of the equation to isolate the term with n : 5 n − 5 p + 5 p = 10 p + 5 p , which simplifies to 5 n = 15 p .
Solving for n Finally, divide both sides of the equation by 5 to solve for n : 5 5 n = 5 15 p , which simplifies to n = 3 p .
Final Answer Therefore, the solution is n = 3 p . Comparing this to the answer choices, we see that it matches option A.
Examples
In physics, if n represents the number of particles and p represents the pressure, the equation n = 3 p indicates a direct proportionality between the number of particles and the pressure. For example, if the pressure is 2 Pascals, then the number of particles is 6. Understanding such relationships is crucial in thermodynamics and statistical mechanics, where the behavior of gases and other systems is analyzed based on pressure, volume, and the number of particles.
The solution to the equation 3 n − 5 p + 2 n = 10 p is n = 3 p . This matches option A in the provided answer choices. Therefore, the answer is n = 3 p .
;
