Solve the equation $6 z+9 z=p z+v$ for $v$.
$v=$
Answer (2)
Combine like terms: 6 z + 9 z = 15 z .
Rewrite the equation: 15 z = p z + v .
Isolate v : v = 15 z − p z .
Factor out z : v = ( 15 − p ) z . The final answer is ( 15 − p ) z .
Explanation
Understanding the Problem We are given the equation 6 z + 9 z = p z + v and asked to solve for v . This means we want to isolate v on one side of the equation.
Combining Like Terms First, combine the like terms on the left side of the equation: 6 z + 9 z = 15 z . So the equation becomes 15 z = p z + v .
Isolating v Next, to isolate v , subtract p z from both sides of the equation: 15 z − p z = p z + v − p z , which simplifies to 15 z − p z = v .
Factoring Out z Finally, we can factor out z from the left side of the equation to get v = ( 15 − p ) z .
Final Answer Therefore, the solution for v is v = ( 15 − p ) z .
Examples
In physics, this type of equation could represent a relationship between forces, masses, and accelerations in a system. For example, if z represents a displacement, p represents a damping coefficient, and v represents a force, the equation describes how the force v is related to the displacement z and the damping force p z . Solving for v allows us to determine the force needed to maintain a certain displacement in the system.
The solution for v in the equation 6 z + 9 z = p z + v is v = ( 15 − p ) z .
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