The problem involves a rectangular flower bed, and you are asked to determine its original length and width based on two scenarios. Let's break it down and solve the system of equations that arises from the two conditions:

1. First scenario:

   • If the width is increased by 2 feet and the length is increased by 3 feet, the area increases by 64 square feet.
   • Let $w$ represent the width and $l$ represent the length.
   • The equation for this scenario is: $(w + 2)(l + 3) = wl + 64$

2. Second scenario:

   • If the width is increased by 3 feet and the length is increased by 2 feet, the area increases by 68 square feet.
   • The equation for this scenario is: $(w + 3)(l + 2) = wl + 68$

Expanding both equations:

• For the first scenario: $(w + 2)(l + 3) = wl + 3w + 2l + 6$ Which simplifies to: $wl + 3w + 2l + 6 = wl + 64$ After canceling out $wl$ on both sides: $3w + 2l + 6 = 64$ Simplifying further: $3w + 2l = 58 \quad \text{(Equation 1)}$

• For the second scenario: $(w + 3)(l + 2) = wl + 2w + 3l + 6$ Which simplifies to: $wl + 2w + 3l + 6 = wl + 68$ After canceling out $wl$ on both sides: $2w + 3l + 6 = 68$ Simplifying further: $2w + 3l = 62 \quad \text{(Equation 2)}$

Solving the system of equations:

We now have the following system of equations:

1. $3w + 2l = 58$
2. $2w + 3l = 62$

To solve, we can use substitution or elimination. I'll use the elimination method.

• Multiply the first equation by 2 and the second equation by 3 to eliminate $l$: $6w + 4l = 116$ $6w + 9l = 186$

• Now subtract the first from the second: $(6w + 9l) - (6w + 4l) = 186 - 116$ This simplifies to: $5l = 70$ So: $l = 14$

• Now substitute $l = 14$ back into one of the original equations, say $3w + 2l = 58$: $3w + 2(14) = 58$ $3w + 28 = 58$ $3w = 30$ $w = 10$

Final Answer:

The width of the flower bed is 10 feet, and the length is 14 feet.

Would you like more details, or do you have any further questions?

Here are 5 related questions to consider:

1. How would the area change if both the width and length were increased by 5 feet?
2. Can you solve this problem using substitution instead of elimination?
3. What is the original area of the rectangular flower bed?
4. How would the problem change if the increases in width and length were switched in both scenarios?
5. How can we use the quadratic equation to check if the solutions are correct?

Tip: When solving systems of equations, always check your solution by substituting the values back into the original equations to ensure accuracy.