From the pair of linear equation for the following problems: The diffe...
Determining the Linear Equations
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
Converting the Equations
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
Solving the Equations
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
Finding the Values of x
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.