Let's start by using algebra to solve for the width and length of the rectangle.
Let x be the width of the rectangle. Then, we know that the length is 1 yard less than twice the width. We can write this as:
length = 2x - 1
We also know that the length of the rectangle is 21 square yards. We can write this as:
length x width = 21
Substituting the expression for length from the first equation into the second equation, we get:
(2x - 1) x x = 21
Simplifying the equation:
2x^2 - x - 21 = 0
Now we can use the quadratic formula to solve for x:
x = (-b ± sqrt(b^2 - 4ac)) / 2a
In this case, a = 2, b = -1, and c = -21. Substituting these values into the quadratic formula, we get:
x = (-(-1) ± sqrt((-1)^2 - 4(2)(-21))) / 2(2)
Simplifying:
x = (1 ± sqrt(169)) / 4
x = (1 ± 13) / 4
x = 3 or x = -7/2
Since the width of a rectangle cannot be negative, we can ignore the negative solution. Therefore, the width of the rectangle is 3 yards.
Using the expression for length from the first equation, we can find the length of the rectangle:
length = 2x - 1
length = 2(3) - 1
length = 5
Therefore, the dimensions of the rectangle are 3 yards by 5 yards.