To solve this problem, we need to first determine the pair of linear equations based on the given information. Let's denote the two numbers as x and y.

1. The difference of two numbers is 4:
This can be expressed as:
x - y = 4

2. The difference of their reciprocal is 4 by 21:
The reciprocal of a number can be obtained by taking the inverse of the number. So, the reciprocal of x is 1/x, and the reciprocal of y is 1/y.
The difference of their reciprocals can be expressed as:
1/x - 1/y = 4/21



To make the equations easier to work with, we can simplify the second equation by finding a common denominator for the fractions:

1. x - y = 4

2. (y - x) / (xy) = 4/21
Multiplying both sides by 21xy to eliminate the fractions:
21(y - x) = 4xy



Now we have a system of linear equations. We can solve them using various methods such as substitution or elimination. Let's use the substitution method:

1. Rearrange the first equation to solve for x:
x = y + 4

2. Substitute this value of x into the second equation:
21(y - (y + 4)) = 4(y + 4)y

Simplifying the equation:
21(-4) = 4(y^2 + 4y)

-84 = 4y^2 + 16y

Rearrange the equation to form a quadratic equation:
4y^2 + 16y + 84 = 0

Divide the equation by 4 to simplify:
y^2 + 4y + 21 = 0

Factorizing the quadratic equation:
(y + 3)(y + 7) = 0

Setting each factor to zero:
y + 3 = 0 or y + 7 = 0

Solving for y:
y = -3 or y = -7



Now that we have the values of y, we can substitute them back into the first equation to find the corresponding values of x:

1. For y = -3:
x = -3 + 4
x = 1

2. For y = -7:
x = -7 + 4
x = -3

Therefore, the two numbers are 1 and -3 (or -3 and 1), which satisfy the given conditions.