To solve for the length [tex]\( l \)[/tex] in terms of the perimeter [tex]\( P \)[/tex] and the width [tex]\( w \)[/tex] of a rectangle, we can manipulate the formula [tex]\( P = 2l + 2w \)[/tex] using algebraic properties. Here’s the step-by-step process:
1. Start with the given formula:
[tex]\[
P = 2l + 2w
\][/tex]
This equation represents the perimeter of a rectangle in terms of its length and width.
2. Subtract [tex]\( 2w \)[/tex] from both sides:
[tex]\[
P - 2w = 2l
\][/tex]
Justification: Using the property of equality (what you do to one side of the equation, you must do to the other) to isolate the term involving [tex]\( l \)[/tex].
3. Divide both sides by 2:
[tex]\[
\frac{P - 2w}{2} = l
\][/tex]
Justification: Using the division property of equality to solve for [tex]\( l \)[/tex]. Dividing both sides by 2 simplifies the equation.
4. Write the final solution:
[tex]\[
l = \frac{P - 2w}{2}
\][/tex]
So, the length [tex]\( l \)[/tex] is given by the formula [tex]\(\frac{P - 2w}{2}\)[/tex] when you know the perimeter [tex]\( P \)[/tex] and the width [tex]\( w \)[/tex] of the rectangle.
