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When solving equations, one of the primary goals is often to isolate the variable we are interested in. In this exercise, we are asked to solve for the variable L in the equation given by: P = 2L + 2W. To isolate L, we need to get it alone on one side of the equation.

Think of isolating a variable like peeling away layers of an onion. You need to remove other terms and factors to leave the variable by itself. In this case, we start by isolating the term containing L, which means we first need to remove the other parts of the equation that are not related to L, specifically the term 2W.

By subtracting 2W from both sides of the equation, we effectively shift all terms involving W to the other side and simplify the equation to: P - 2W = 2L.

This step helps to clear out additional variables and put us closer to finding our desired variable.

Algebraic manipulation involves performing operations that simplify an equation and make it possible to solve for the desired variable. This often includes addition, subtraction, multiplication, division, and sometimes more complex operations.

In our given problem, after isolating 2L on one side, we need to get L by itself. Starting with the equation P - 2W = 2L, we observe that L is multiplied by 2. To isolate L completely, we divide every term by 2.

Remember, whatever you do to one side of the equation, you must also do to the other side to maintain balance. So, dividing both sides by 2 gives us: \( \frac{P - 2W}{2} = L \).

This division step simplifies to L = (P - 2W) / 2, completing the process of solving for L while maintaining the equality of the equation.

Linear equations are equations in which the highest power of any variable is 1. This means the equation graphs as a straight line and includes terms like x, 3y, or -5z, where the variables don't have exponents higher than 1.

The given problem \( P = 2L + 2W \) is a linear equation because both L and W are to the first power. This makes it relatively straightforward to solve using basic algebraic techniques.

Understanding linear equations is crucial because they form the foundation for more complex algebraic problems and real-world applications, like calculating distances, budgeting, and more.

Working through our example, we began with a linear equation and performed algebraic steps to isolate L and find its value in terms of P and W. This process highlights the simplicity and elegance of linear equations as we can solve them with systematic and logical steps.