The sum of two integers is 12. Rhe difference in their reciprocal is 2/9. Determine the 2 integers.
If y is the larger number
x+y = 12
1/x - 1/y = 2/9
1/x- 1/(12-x) = 2/9
Now it ends much nicer
thanks, oobleck
I should really read the questions more carefully.
To solve this problem, let's assume the two integers are x and y.
According to the given information, we have two equations:
1. x + y = 12 (Equation 1: the sum of two integers is 12)
2. 1/x - 1/y = 2/9 (Equation 2: the difference in their reciprocal is 2/9)
Now, let's solve these equations step by step:
Step 1: Rearrange Equation 2.
Multiply both sides of Equation 2 by xy to eliminate the fractions:
(xy)(1/x - 1/y) = (xy)(2/9)
This simplifies to:
y - x = (2/9)xy
Step 2: Simplify Equation 2.
Rearrange Equation 2 by moving all terms to one side:
(2/9)xy - y + x = 0
Step 3: Solve for x in terms of y.
Let's solve Equation 3 for x:
x = (9y - 2xy)/(9)
Step 4: Substitute x in Equation 1.
Substitute the value of x from Equation 3 into Equation 1:
(9y - 2xy)/(9) + y = 12
Simplify this equation:
Multiply both sides by 9:
9y - 2xy + 9y = 108
Combine like terms:
18y - 2xy = 108
Step 5: Solve for y.
Rearrange Equation 5 to solve for y:
18y = 108 + 2xy
18y = 108 + 2x(12 - y)
18y = 108 + 24x - 2xy
18y + 2xy = 108 + 24x
Factor out y:
y(18 + 2x) = 108 + 24x
y = (108 + 24x)/(18 + 2x)
Step 6: Substitute y value.
Now, substitute the value of y back into Equation 3 to solve for x:
x = (9[(108 + 24x)/(18 + 2x)] - 2x[(108 + 24x)/(18 + 2x)])/(9)
Simplify Equation 6:
x = (972 + 216x - 216x - 48x^2)/(162 + 18x)
x = (972)/(162 + 18x)
x = 972/(18(9 + x))
x = 972/(162 + 18x)
Now, we have an equation solely in terms of x.
Step 7: Solve for x.
Multiply both sides by 162 + 18x to eliminate the denominator:
x(162 + 18x) = 972
Simplify this equation:
18x^2 + 162x - 972 = 0
Divide through by 18 to simplify:
x^2 + 9x - 54 = 0
Factorize this quadratic equation:
(x + 12)(x - 3) = 0
This gives us two possible values for x: x = -12 or x = 3.
Step 8: Substitute x in Equation 1.
Substitute each value of x into Equation 1 to find the respective values of y:
For x = -12:
(-12) + y = 12
y = 12 + 12
y = 24
For x = 3:
3 + y = 12
y = 12 - 3
y = 9
Therefore, the two integers are (-12, 24) and (3, 9).
one number --- x
the other ----- y
x+y=12 -----> x = 12-y
x - 1/y = 2/9
12-y - 1/y = 2/9
multiply by 9y
108y - 9y^2 - 9 = 2y
9y^2 - 106y + 9 = 0
solve the quadratic, does not come out nice