Rectangle Perimeter Problem: Solve By Substitution

Hey guys! Let's break down this rectangle problem step by step using the substitution method. This problem involves finding the dimensions of a rectangle given its perimeter and a relationship between its length and width. We'll walk through each step, making it super easy to follow. So, let's dive in!

1. Defining the Variables

First, we need to define our variables. Let's use:

• l = length of the rectangle
• w = width of the rectangle

Clear and straightforward, right? This sets the stage for translating the word problem into mathematical equations. Properly defining variables is crucial for setting up the equations correctly. This step ensures that we understand what each variable represents in the context of the problem. Without this clarity, the subsequent steps might become confusing, leading to errors in the solution. Always take a moment to clearly define your variables before proceeding further!

Knowing what each variable stands for helps in interpreting the final results and verifying whether they make sense in the original problem context. For example, if we find a negative value for the width, it indicates an error in our calculations since dimensions cannot be negative. By defining the variables explicitly, we lay a solid foundation for the rest of the solution process, ensuring accuracy and clarity throughout.

By establishing these variables, we're setting ourselves up for success in translating the word problem into a solvable system of equations. Trust me, it makes the whole process way smoother!

2. Setting Up the Equations

Now, let's translate the given information into equations.

• Perimeter: The perimeter of a rectangle is given by 2l + 2w. We know the perimeter is 50 cm, so we have: 2l + 2w = 50
• Length vs. Width: The length exceeds the width by 5 cm, which means: l = w + 5

So, we have a system of two equations:

1. 2l + 2w = 50
2. l = w + 5

These equations perfectly capture the relationships described in the problem. This is like translating from English to Math! The first equation represents the total perimeter of the rectangle, while the second equation describes the relationship between the length and width. Together, they form a system that we can solve to find the unknown dimensions of the rectangle. Accurately setting up these equations is paramount because the entire solution hinges on them. A mistake here can lead to incorrect answers, so it's always a good idea to double-check that they accurately reflect the information given in the problem.

By establishing these equations, we've effectively created a mathematical model of the problem. Now, all that's left is to solve this model using the substitution method, which we'll tackle in the next section. We’re on our way to finding the exact dimensions of the rectangle, making the problem tangible and solvable.

3. Solving by Substitution

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. Since we already have l isolated in the second equation (l = w + 5), we can substitute it into the first equation:

2(w + 5) + 2w = 50

Now, let's simplify and solve for w:

2w + 10 + 2w = 50 4w + 10 = 50 4w = 40 w = 10

Great! We found that the width, w, is 10 cm. Now, we can use this value to find the length, l.

This is where the magic of substitution shines! By replacing l with its equivalent expression in terms of w, we've transformed a two-variable equation into a single-variable equation. This simplification is key to unlocking the solution. The algebraic manipulations in this step, such as distributing, combining like terms, and isolating w, are fundamental skills in solving equations. It's essential to perform these steps accurately to avoid errors that would propagate through the rest of the solution. Double-checking each step of the simplification process ensures that we arrive at the correct value for w.

Once we have the value of w, we can easily substitute it back into the equation l = w + 5 to find the length. This highlights the elegance and efficiency of the substitution method, allowing us to solve for each variable in a systematic and straightforward manner. With w determined, we're one step closer to completely characterizing the dimensions of the rectangle and answering the original problem.

4. Finding the Length

Now that we know w = 10, we can find l using the equation l = w + 5:

l = 10 + 5 l = 15

So, the length, l, is 15 cm.

Finding the length after determining the width is a crucial step in completing the problem. Substituting the value of w into the equation l = w + 5 is a direct application of the relationship defined in the problem statement. This calculation is simple but essential, as it provides the second dimension needed to fully describe the rectangle. The ease with which we can find l after finding w demonstrates the effectiveness of the substitution method in simplifying the solution process. Ensuring the accuracy of this calculation is important, as it directly impacts the final answer and the subsequent verification step.

With both the length and width now determined, we have a complete set of dimensions for the rectangle. This allows us to proceed to the next step, which is to verify our solution to ensure that it satisfies the original conditions of the problem. This verification step is a crucial check that confirms the correctness of our calculations and the validity of our solution.

5. Verification

To verify our solution, we need to check if the values we found satisfy both original equations:

• Perimeter: 2l + 2w = 50 2(15) + 2(10) = 30 + 20 = 50 (Correct!)
• Length vs. Width: l = w + 5 15 = 10 + 5 (Correct!)

Both equations are satisfied, so our solution is correct!

Verification is an indispensable part of the problem-solving process. By substituting our calculated values for l and w back into the original equations, we're essentially testing whether our solution is consistent with the information provided in the problem statement. This step helps catch any errors that may have occurred during the algebraic manipulations or calculations. Confirming that the values satisfy both the perimeter equation and the length-width relationship gives us confidence that our solution is indeed correct. Without this verification, we would be unsure if our answer is valid, potentially leading to incorrect conclusions. This process not only validates the solution but also reinforces understanding of the problem and the relationships between the variables.

By rigorously verifying our solution, we can confidently move on to stating the final answer, knowing that it is accurate and consistent with the problem's requirements. This step ensures that we provide a reliable and correct solution to the question at hand, demonstrating a thorough understanding of the problem-solving process.

6. Final Answer

The length of the rectangle is 15 cm, and the width is 10 cm.

So, there you have it! We successfully solved the rectangle perimeter problem using the substitution method. Wasn't that fun? Keep practicing, and you'll become a math whiz in no time!

The final answer is the culmination of all the steps taken throughout the problem-solving process. Clearly stating the values of the length and width, along with their units (cm), provides a complete and unambiguous solution to the original question. This final step ensures that the answer is easily understandable and can be directly applied to the context of the problem. Presenting the answer in a clear and concise manner is essential for effective communication of the solution. This allows anyone reading the solution to quickly grasp the results and understand the dimensions of the rectangle in question. By providing a well-defined final answer, we conclude the problem-solving process in a satisfactory and conclusive manner.