The length of a rectangle is 2 cm more than three times the width. Find the dimensions of this rectangle if the area is 21 cm^2.

I know the answer is 9 & 2 1/3 but I found it by trial and error. Is there an easier way to find the answer to this question? Thanks!

the answer can be found by using algebra.

L = 3W + 2
L*W = 21
(3W+2)*W = 21
3w^2 + 2W -21 = 0
(3W -7)(W +3) = 0
W = 7/3
Ignore the negative root

Yes, there is a systematic way to find the dimensions of the rectangle without relying on trial and error. Let's break down the problem step by step:

1. Understand the problem:
We are given that the length of the rectangle is 2 cm more than three times the width, and the area is 21 cm^2. We need to find the dimensions of the rectangle.

2. Define variables:
Let's represent the width of the rectangle as 'w' cm. Since the length is 2 cm more than three times the width, we can represent the length as (3w + 2) cm.

3. Use the area formula:
The area of a rectangle is given by the formula: Area = length x width.
Substitute the values into the formula: 21 = (3w + 2) x w.

4. Solve the equation:
Expand the equation: 21 = 3w^2 + 2w.
Rearrange the equation to bring it to standard quadratic form: 3w^2 + 2w - 21 = 0.
Since this equation is quadratic, we can solve it by factoring, completing the square, or using the quadratic formula. In this case, factoring might be the most convenient approach.

5. Factor the quadratic equation:
Write the equation in factored form: (w - 3)(3w + 7) = 0.
Set each factor equal to zero:
w - 3 = 0 or 3w + 7 = 0.

6. Solve for 'w':
For the first factor:
w - 3 = 0
Adding 3 to both sides:
w = 3.

For the second factor:
3w + 7 = 0
Subtracting 7 from both sides:
3w = -7
Dividing by 3:
w = -7/3.

Since the width of a rectangle cannot be negative, our valid solution is w = 3 cm.

7. Calculate the length:
Using the value of w obtained above, substitute it back into the expression for the length:
length = 3w + 2 = 3(3) + 2 = 9 + 2 = 11 cm.

Therefore, the dimensions of the rectangle are width = 3 cm and length = 11 cm.

Please note that the answer you provided (9 and 2 1/3) does not satisfy the given area of 21 cm^2, so it seems to be incorrect.