91Ó°ÊÓ

Solving equations is a cornerstone of algebra. It involves finding values of variables that make an equation true. For example, in our exercise, we began with the equation \( r = \sqrt{\frac{M m}{F}} \). Our task was to find the value of \( F \). Solving equations like these usually involves several steps:
1. **Understanding the Equation**: Know which variable you're solving for and what each term in the equation represents.
2. **Step-by-Step Manipulation**: Follow logical steps, like squaring both sides to eliminate square roots, or multiplying/dividing to isolate terms.
In this case, after squaring both sides, we worked our way to: \( r^2 = \frac{M m}{F} \). From here, the goal is to manipulate the equation until the desired variable (\( F \)) stands alone.

Isolating the variable is a key skill in algebra. It means rearranging the equation so that the variable you are solving for stands alone on one side of the equation. In our example, we wanted to isolate \( F \) in the equation \( r^2 = \frac{M m}{F} \). The steps to do this are:
1. **Eliminate Denominators**: Multiplying both sides by \( F \) to get \( F r^2 = M m \).
2. **Solve for the Variable**: Dividing both sides by \( r^2 \) results in \( F \) being isolated: \( F = \frac{M m}{r^2} \).
These steps show how algebraic manipulation aids in isolating the desired variable.

Algebraic manipulation refers to the processes used to simplify expressions or solve equations. Techniques include adding, subtracting, multiplying, or dividing both sides of an equation, and applying functions like squaring or taking roots. In our exercise, we used:
1. **Squaring Both Sides**: To remove the square root from \( r = \sqrt{\frac{M m}{F}} \), giving us \( r^2 = \frac{M m}{F} \).
2. **Multiplication**: To clear a fraction, we multiplied both sides by \( F \), resulting in \( F r^2 = M m \).
3. **Division**: Finally, dividing both sides by \( r^2 \) isolated our variable of interest, leading to \( F = \frac{M m}{r^2} \).
By applying these steps, you can manipulate almost any algebraic equation to solve for any variable.