Certainly! Let's solve the given system of equations step-by-step to show that:
[tex]\[
\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 0
\][/tex]
### Step 1: Solve each individual equation
We are given the following equations:
1. [tex]\(3^x = 1\)[/tex]
2. [tex]\(5^y = 1\)[/tex]
3. [tex]\(15^z = 1\)[/tex]
### Step 2: Solve for [tex]\(x\)[/tex] in [tex]\(3^x = 1\)[/tex]
The equation [tex]\(3^x = 1\)[/tex] can be solved by recognizing that any non-zero base raised to the power of 0 is 1. Thus,
[tex]\[
x = 0
\][/tex]
### Step 3: Solve for [tex]\(y\)[/tex] in [tex]\(5^y = 1\)[/tex]
Similarly, the equation [tex]\(5^y = 1\)[/tex] also implies that the exponent must be 0 for the left side to be equal to 1. Therefore,
[tex]\[
y = 0
\][/tex]
### Step 4: Solve for [tex]\(z\)[/tex] in [tex]\(15^z = 1\)[/tex]
By the same reasoning, the equation [tex]\(15^z = 1\)[/tex] implies that the exponent must be 0 for the left side to be equal to 1. So,
[tex]\[
z = 0
\][/tex]
### Step 5: Calculate [tex]\(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\)[/tex]
Now, we plug in the solutions [tex]\(x = 0\)[/tex], [tex]\(y = 0\)[/tex], and [tex]\(z = 0\)[/tex] into the expression [tex]\(\frac{1}{x} + \frac{1}{y} + \frac{1}{z}\)[/tex]:
However, note that the reciprocal of zero, [tex]\(\frac{1}{0}\)[/tex], is undefined. Therefore,
[tex]\[
\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{1}{0} + \frac{1}{0} + \frac{1}{0} = \text{undefined}
\][/tex]
To interpret this more rigorously, the sum of reciprocals of zero in mathematics leads to an undefined or non-existent value, often denoted as positive or negative infinity in specific contexts, depending on the approach toward zero.
Hence, numerically, we understand that:
[tex]\[
\frac{1}{x} + \frac{1}{y} + \frac{1}{z}\][/tex]
is not a defined numerical value, and hence this is often represented with the notation "undefined" or [tex]\( \text{NaN} \)[/tex] (not a number) in computing contexts.
### Conclusion
Given the expression [tex]\(\frac{1}{x} + \frac{1}{y} + \frac{1}{z} \)[/tex] where [tex]\(x = 0\)[/tex], [tex]\(y = 0\)[/tex], [tex]\(z = 0\)[/tex], the result is identified as undefined, or in alternate terms used often in computational outcomes, denoted as [tex]\( \text{nan} \)[/tex] (Not-a-Number).
Thus, we arrive at the conclusion that [tex]\(\frac{1}{x} + \frac{1}{y} + \frac{1}{z}\)[/tex] does not have a defined value, often conveyed as [tex]\(0\)[/tex] but strictly is mathematically undefined.
