Solving Rectangle Dimensions Based on Perimeter and Length

Understanding the dimensions of a rectangle can often involve solving a series of equations, especially when given the relationship between the length and the width as well as the perimeter. In this article, we'll explore how to find the dimensions of a rectangle when we know that its length is 6 inches less than twice its width and that its perimeter is 72 inches. This problem requires the application of algebraic principles to find the dimensions accurately.

Formulating the Equations

Let's denote the width of the rectangle as W (in inches) and the length of the rectangle as L (in inches).

Step 1: Expressing the Length in Terms of the Width

From the problem statement, we know that the length is 6 inches less than twice the width. This can be written as:

L 2W - 6

Step 2: Understanding the Perimeter

The perimeter of a rectangle is given by the formula:

P 2L 2W

Given that the perimeter is 72 inches, we can write the equation as:

2L 2W 72

Solving the System of Equations

We now have two equations:

L 2W - 6 2L 2W 72

To solve for the dimensions, we can substitute the first equation into the second equation.

Substitution and Simplification

Substituting L 2W - 6 into the perimeter equation:

2(2W - 6) 2W 72

Simplifying this equation:

4W - 12 2W 72

6W - 12 72

6W 84

W 14

Now that we have the width, we can find the length by substituting W 14 back into the first equation:

L 2(14) - 6 28 - 6 22

The dimensions of the rectangle are:

Width: 14 inches Length: 22 inches

Conclusion

This method of solving for the dimensions of a rectangle using a system of equations is a valuable skill in algebra and can be applied to various real-world scenarios. Understanding how to set up and solve these equations with clear and concise steps is crucial for anyone working with geometric shapes and their properties.

Related Topics and Further Exploration

For those interested in further exploring the topic, the following related concepts are worth considering:

Perimeter Equations: Understanding how the perimeter of different shapes (like triangles, trapezoids, and circles) is calculated. System of Equations: Solving more complex systems involving multiple variables and equations. Algebraic Manipulation: Techniques for simplifying and solving algebraic expressions and equations.