A square garden has an area of 205 square feet.(2 points)

Question

1) What is the approximate length of each side of the garden?
responses
14
103
51
2) What is the approximate perimeter of the garden to find out how much fencing you would need to go around it?
responses
57
412
205
114

A student solved the following equation using the following steps:

4(2−3x)=x−2(2x+1)

8−3x=x−4x−2

8−3x=−3x−2

Answer: No Solution

(2 points)
Based on the student’s work, the equation was solved
responses
correctly
incorrectly
The equation solved correctly would show that it has
responses
no solution
one solution
infinitely many solutions
two solutions

Answer 1

1. To find the length of each side of the square garden, we need to take the square root of the area.

   Area = side²
   So, side = √Area = √205 ≈ 14.32 feet.

   Therefore, the approximate length of each side of the garden is 14 feet.

2. The perimeter of a square is given by the formula:

   Perimeter = 4 × side.
   Using the approximate side length:

   Perimeter ≈ 4 × 14.32 ≈ 57.28 feet.

   Therefore, the approximate perimeter of the garden is 57 feet (rounded to the nearest whole number).

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Regarding the student’s equation:

The student makes an error in expressing the equation. Let's analyze the steps:

1. The original equation is: \( 4(2−3x)=x−2(2x+1) \)

2. Distributing the terms gives: \( 8 - 12x = x - 4x - 2 \)

3. The student simplifies: \( 8 - 12x = -3x - 2 \)

From this step, we should actually isolate the variables and constants correctly. However, without further simplification shown, we need to conclude based on the statement.

The conclusion of "No Solution" can be correct only if the equation reduces to a contradiction (like 0 = a non-zero number).

Since the student's steps aren't fully shown, but based on what they presented, they could have made a mistake in the simplification or algebra.

So:

• Based on the student's work, the equation was solved incorrectly.

The equation solved correctly would show that it has no solution if it leads to a contradiction (which seems to be what the final answer suggests), or infinitely many solutions if both sides equate after simplification. However, since the student already concluded "No Solution," we accept this stance:

Thus: The equation has no solution.