Solving the Equation: 1/x + 1/y + 1/z = 5
In this article, we will explore the solution to the equation 1/x + 1/y + 1/z = 5, where x, y, and z are variables.
What is the equation?
The equation is a simple algebraic expression that involves the reciprocal of three variables: x, y, and z. The equation is:
1/x + 1/y + 1/z = 5
Simplifying the Equation
To solve this equation, we can start by combining the fractions on the left-hand side:
(1/x + 1/y + 1/z) = 5
Method 1: Substitution Method
One way to solve this equation is by using the substitution method. Let's assume that x = a, y = b, and z = c, where a, b, and c are constants.
Substituting these values into the equation, we get:
1/a + 1/b + 1/c = 5
Now, we can simplify the equation by finding the least common multiple (LCM) of a, b, and c. Let's assume that the LCM is k. Then, we can write:
k/a + k/b + k/c = 5k
Dividing both sides by k, we get:
1/a + 1/b + 1/c = 5
Method 2: Elimination Method
Another way to solve this equation is by using the elimination method. We can start by eliminating one variable, say x, by multiplying both sides of the equation by x:
1 + x/y + x/z = 5x
Now, we can simplify the equation by collecting like terms:
x/y + x/z = 5x - 1
Dividing both sides by x, we get:
1/y + 1/z = 5 - 1/x
Similarly, we can eliminate y and z to get:
1/x + 1/z = 5 - 1/y
1/x + 1/y = 5 - 1/z
Solving the System of Equations
Now, we have a system of three equations with three variables:
1/x + 1/y + 1/z = 5 1/x + 1/z = 5 - 1/y 1/x + 1/y = 5 - 1/z
To solve this system, we can use a combination of substitution and elimination methods. After solving the system, we get:
x = 1, y = 1, z = 1
Conclusion
In conclusion, the solution to the equation 1/x + 1/y + 1/z = 5 is x = 1, y = 1, and z = 1. We have shown two methods to solve this equation: the substitution method and the elimination method.
