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To solve for a variable means to isolate that variable on one side of an equation.
This involves using various mathematical operations to 'undo' the operations surrounding that variable.
For instance, in the given problem, we start with the equation that contains the variable we're interested in, which is M.
\r\rWhen solving for M in the equation \r\r\(r = \sqrt{\frac{M m}{F}}\), our goal is to isolate M so that it stands alone on one side of the equation.
We can employ arithmetic operations and algebraic techniques to achieve this.
\r\rThis process can include squaring both sides, adding or subtracting terms, and multiplying or dividing both sides by a number or another variable.
\rIn each step, we perform operations that gradually simplify the equation and help us isolate the variable we are solving for.

Algebraic manipulation is the core technique used when solving equations.
It consists of performing operations to both sides of the equation to maintain equality.
This helps to transform the equation into a simpler or more useful form.\r\rIn our problem, we carried out algebraic manipulation in several steps:\r\r

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• First, we squared both sides of the equation \(r = \sqrt{\frac{M m}{F}}\) to remove the square root: \(r^2 = \frac{M m}{F}\).
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• Next, we multiplied both sides by F to move F to the left side: \(F r^2 = M m\).
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• Finally, we divided both sides by m to isolate M: \(M = \frac{F r^2}{m}\).
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Understanding each operation helps to see how we reverse the original operations applied to the variable to isolate it.
This allows us to grasp how the variable relates to the other terms in the equation.

Formula rearrangement is a specific application of algebraic manipulation.
It entails changing the arrangement of a formula to solve for a specific variable or to express the formula in a different perspective.
\r\rFor the given problem, rearranging the formula involves starting with the original formula \(r = \sqrt{\frac{M m}{F}}\) and following a sequence of steps to solve for M.\r\r

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• First, we squared both sides to eliminate the square root, rendering the equation in a simpler quadratic form.
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• We then multiplied by F to shift it across the equality, aiding in isolating M on one side.
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• Finally, we divided by m, ensuring that M is separated and expressed as \(M = \frac{F r^2}{m}\).
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\rRewriting the formula like this provides a new way to understand how different variables in the original equation are related.
Each step during formula rearrangement is guided by the algebraic principles that maintain the balance of the equation.
This is crucial to ensure accuracy and consistency in mathematical problem-solving.