Hey guys! Ever found yourself staring at a formula and feeling like it's written in another language? Don't worry, we've all been there. Today, we're going to break down a common formula – the perimeter of a rectangle – and learn how to solve for one of its variables, specifically the length (l). We'll take a simple, step-by-step approach, so even if math isn't your favorite subject, you'll be able to follow along and master this skill.
Understanding the Perimeter Formula
Before we dive into the steps, let's quickly recap the perimeter formula itself. The perimeter (P) of a rectangle is the total distance around its outside. Imagine you're building a fence around a rectangular yard; the perimeter is the total length of fencing you'll need. The formula that represents this is:
P = 2l + 2w
Where:
- P represents the perimeter.
- l represents the length of the rectangle.
- w represents the width of the rectangle.
This formula tells us that the perimeter is equal to twice the length plus twice the width. This makes sense because a rectangle has two sides of equal length and two sides of equal width. Now that we understand the formula, let's get into the steps for solving for l.
Step-by-Step Solution: Isolating the Length (l)
Our goal here is to get l by itself on one side of the equation. This is called isolating the variable. We'll do this by using inverse operations – operations that "undo" each other. Think of it like unwrapping a present; you need to reverse the steps that were taken to wrap it.
Step 1: Start with the Original Formula
The first thing we always do is write down the formula we're working with. This helps keep us organized and ensures we don't miss any steps. So, let's write it down:
P = 2l + 2w
This is our starting point. We know P (the perimeter) and w (the width), and we want to find l (the length).
Step 2: Subtract 2w from Both Sides
Remember, in algebra, whatever we do to one side of the equation, we must do to the other side to keep the equation balanced. Our goal is to isolate the term with l (which is 2l), so we need to get rid of the 2w term. Since 2w is being added, we'll use the inverse operation – subtraction. We subtract 2w from both sides of the equation:
P - 2w = 2l + 2w - 2w
Notice that on the right side, +2w and -2w cancel each other out, leaving us with:
P - 2w = 2l
We're one step closer to isolating l! We've successfully moved the 2w term to the other side of the equation.
Step 3: Divide Both Sides by 2
Now we have P - 2w = 2l. The l is being multiplied by 2, so to isolate l, we need to undo this multiplication. The inverse operation of multiplication is division. We'll divide both sides of the equation by 2:
(P - 2w) / 2 = (2l) / 2
On the right side, the 2 in the numerator and the 2 in the denominator cancel each other out, leaving us with:
(P - 2w) / 2 = l
We did it! We've successfully isolated l. Now we have an equation that tells us exactly how to calculate the length if we know the perimeter and the width.
Step 4: Rewrite the Equation (Optional)
While (P - 2w) / 2 = l is perfectly correct, it's often more common to see the variable we're solving for on the left side of the equation. We can easily rewrite the equation to switch the sides without changing its meaning:
l = (P - 2w) / 2
This is the final form of the equation, and it clearly shows how to calculate the length (l) given the perimeter (P) and the width (w).
Putting It All Together: The Steps in Order
Let's recap the steps we took to solve for l in the perimeter formula:
- Start with the original formula: P = 2l + 2w
- Subtract 2w from both sides: P - 2w = 2l
- Divide both sides by 2: (P - 2w) / 2 = l
- (Optional) Rewrite the equation: l = (P - 2w) / 2
By following these steps, you can confidently solve for the length of a rectangle when you know its perimeter and width. Remember, the key is to use inverse operations to isolate the variable you're trying to find.
Example Time: Let's Use the Formula
Okay, now that we know the steps, let's put them into action with an example. Imagine we have a rectangular garden with a perimeter of 50 feet and a width of 10 feet. We want to find the length of the garden. Let's use our formula:
l = (P - 2w) / 2
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Substitute the values: We know P = 50 feet and w = 10 feet, so we plug these values into the formula:
l = (50 - 2 * 10) / 2
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Simplify the expression: Now we follow the order of operations (PEMDAS/BODMAS). First, we multiply 2 * 10, which equals 20:
l = (50 - 20) / 2
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Continue simplifying: Next, we subtract 20 from 50, which equals 30:
l = 30 / 2
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Final step: Divide: Finally, we divide 30 by 2, which gives us 15:
l = 15
So, the length of the garden is 15 feet. See? It's not so scary once you break it down into steps!
Why is This Important? Real-World Applications
You might be thinking,
