To find an equivalent equation for the perimeter of a rectangle given by Paolo:
[tex]\[ P = 2(l + w) \][/tex]
we need to isolate [tex]\( w \)[/tex] in terms of [tex]\( P \)[/tex] and [tex]\( l \)[/tex]. Here’s the step-by-step solution:
1. Start with the given equation:
[tex]\[ P = 2(l + w) \][/tex]
2. Divide both sides by 2 to simplify the equation:
[tex]\[ \frac{P}{2} = l + w \][/tex]
3. Isolate [tex]\( w \)[/tex] by subtracting [tex]\( l \)[/tex] from both sides of the equation:
[tex]\[ \frac{P}{2} - l = w \][/tex]
4. Rewrite the equation to make it clearer:
[tex]\[ w = \frac{P}{2} - l \][/tex]
We now have [tex]\( w \)[/tex] isolated in terms of [tex]\( P \)[/tex] and [tex]\( l \)[/tex]:
[tex]\[ w = \frac{P}{2} - l \][/tex]
To match this with the given answer choices, let’s manipulate the expression by eliminating the fraction:
5. Start with the equation [tex]\( w = \frac{P}{2} - l \)[/tex]:
[tex]\[ w = \frac{P}{2} - l \][/tex]
6. Multiply both sides of the equation by 2 to clear the fraction:
[tex]\[ 2w = P - 2l \][/tex]
7. Divide both sides by 2 to make it match the given choices:
[tex]\[ w = \frac{P - 2l}{2} \][/tex]
So, the equation equivalent to [tex]\( P = 2(l + w) \)[/tex] is:
[tex]\[ w = \frac{P - 2l}{2} \][/tex]
Therefore, the correct answer from the given choices is:
[tex]\[ w = \frac{P - 2l}{2} \][/tex]
