Solve the following formula for $b$
$\begin{array}{l}
y=3(5 a+7 b) \\
b=\square
\end{array}$
Answer (1)
Distribute the constant: y = 15 a + 21 b .
Isolate the term with b : y − 15 a = 21 b .
Divide to solve for b : b = 21 y − 15 a .
The solution for b is: b = 21 y − 15 a .
Explanation
Understanding the Problem We are given the equation y = 3 ( 5 a + 7 b ) and our goal is to solve for b , which means we want to isolate b on one side of the equation.
Distributing the Constant First, distribute the 3 on the right side of the equation:
y = 3 ( 5 a ) + 3 ( 7 b )
y = 15 a + 21 b
Isolating the Term with b Next, we want to isolate the term with b . To do this, subtract 15 a from both sides of the equation:
y − 15 a = 15 a + 21 b − 15 a
y − 15 a = 21 b
Solving for b Finally, to solve for b , divide both sides of the equation by 21 :
21 y − 15 a = 21 21 b
21 y − 15 a = b
So, b = 21 y − 15 a
Final Answer Therefore, the solution for b is:
b = 21 y − 15 a
Examples
In physics, this type of equation can represent relationships between variables in a system. For example, if y represents the total energy, a represents a known constant related to potential energy, and b represents a variable related to kinetic energy, solving for b allows you to determine the kinetic energy component based on the total energy and the known potential energy constant. This is useful in analyzing energy distribution in physical systems.
