Solving the Equation 1/x - 1/y + 2/z = 9
In this article, we will explore the solution to the equation 1/x - 1/y + 2/z = 9. This equation involves three variables, x, y, and z, and we will use algebraic methods to solve for these variables.
Simplifying the Equation
To start, let's simplify the equation by combining the fractions on the left-hand side:
1/x - 1/y + 2/z = (y - x)/(xy) + 2/z = 9
Isolating Variables
To isolate one of the variables, let's start by multiplying both sides of the equation by xyz, the least common multiple of x, y, and z:
xyz(y - x)/xy + 2xz = 9xyz
Simplifying the equation further, we get:
yz - xz + 2xz = 9xyz
Solving for x
Now, let's solve for x. We can start by grouping the terms with x on the left-hand side:
(yz - 3yz)x = yz - 2xz
Factoring out x, we get:
x(yz - 3yz) = yz - 2xz
x = (yz - 2xz)/(yz - 3yz)
Solving for y and z
To solve for y and z, we can use substitution or elimination methods. Since the equation is already complicated, we'll use the substitution method.
Let's assume z = kx, where k is a constant. Substituting this into the equation, we get:
x(y - x + 2kx)/xy = 9x
Simplifying the equation, we get:
y - x + 2kx = 9x
Rearranging the terms, we get:
y = x(8 - 2k)
Now, substituting this expression for y into the original equation, we get:
1/x - 1/x(8 - 2k) + 2/kx = 9
Simplifying the equation, we get:
k = 1/3
Substituting k back into the expression for y, we get:
y = x(8 - 2/3)
y = 22x/3
Conclusion
In conclusion, we have solved the equation 1/x - 1/y + 2/z = 9 and expressed x, y, and z in terms of each other. The solutions are x = (yz - 2xz)/(yz - 3yz), y = 22x/3, and z = x/3. These solutions can be used to find the values of x, y, and z given specific conditions or constraints.
