Let's break the problem down step by step.

Part 1: Assign Variables

• Let $x$ represent the length of the field (in feet).
• Let $y$ represent the width of the field (in feet).

Part 2: Understanding the Perimeter Formula

The formula for the perimeter $P$ of a rectangle is: $P = 2x + 2y$ Where:

• $x$ is the length.
• $y$ is the width.

Part 3: Given Information

• The perimeter is 116 feet, so: $116 = 2x + 2y$
• The width is 24 feet shorter than the length, so: $y = x - 24$

Part 4: Write Two Equations

Using the perimeter formula and the relationship between the length and width, we can write the following system of equations:

1. Perimeter equation: $2x + 2y = 116$
2. Width equation (from the given information): $y = x - 24$

Now, let's substitute the second equation into the first equation.

Step 1: Substitute $y = x - 24$ into the perimeter equation:

$2x + 2(x - 24) = 116$

Step 2: Simplify the equation:

$2x + 2x - 48 = 116$ $4x - 48 = 116$

Step 3: Solve for $x$:

$4x = 116 + 48$ $4x = 164$ $x = \frac{164}{4} = 41$

So, the length $x = 41$ feet.

Step 4: Solve for $y$:

Now, substitute $x = 41$ into the second equation $y = x - 24$: $y = 41 - 24 = 17$

Thus, the width $y = 17$ feet.

Final Answer:

• Length of the field = 41 feet
• Width of the field = 17 feet

Would you like further details or assistance with another problem?

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Here are some related questions that may help expand your understanding:

1. How would you solve for the perimeter if the length and width were given as different expressions (e.g., $x + 2y = 130$)?
2. What would happen to the perimeter if the width were increased by 10 feet?
3. How can you derive the perimeter formula using algebraic expressions for a rectangle?
4. How can the equation system be solved by substitution instead of elimination?
5. Can you generalize the solution method for other geometric shapes?

Tip: When working with word problems involving geometric shapes, always write down the formulas first and identify the relationships between the given quantities. This will help you set up the right system of equations.