oftermine whether each equation represents an inverse proportion. It so, find the constant of proportionality. $$ \frac{1}{20}=x=\frac{5}{y} $$ $$ y+\frac{1}{7} x=\frac{1}{2} $$ $$ 0,1 x=\frac{5}{y} $$

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Let's analyze each equation to determine whether it represents an inverse proportion. If it does, we'll identify the constant of proportionality.

### 1. $$ \frac{1}{20} = x = \frac{5}{y} $$

**Interpretation:**
The equation can be interpreted as:
$$ x = \frac{1}{20} $$
and
$$ x = \frac{5}{y} $$

**Analysis:**
Setting the two expressions for $$ x $$ equal to each other:
$$ \frac{1}{20} = \frac{5}{y} $$

Cross-multiplying to solve for $$ y $$:
$$ y = 5 \times 20 = 100 $$

**Inverse Proportion Check:**
The general form of an inverse proportion is:
$$ x = \frac{k}{y} $$
where $$ k $$ is the constant of proportionality.

Here, comparing $$ x = \frac{5}{y} $$ with the general form, we see that:
$$ k = 5 $$

**Conclusion:**
This equation **does** represent an inverse proportion with a constant of proportionality $$ k = 5 $$.

---

### 2. $$ y + \frac{1}{7}x = \frac{1}{2} $$

**Interpretation:**
This is a linear equation involving both $$ y $$ and $$ x $$.

**Rearrangement:**
Solving for $$ y $$:
$$ y = -\frac{1}{7}x + \frac{1}{2} $$

**Inverse Proportion Check:**
Inverse proportions require that one variable is proportional to the reciprocal of the other, typically in the form:
$$ y = \frac{k}{x} $$
or
$$ x = \frac{k}{y} $$

The given equation is a linear relationship and does not fit the form of an inverse proportion.

**Conclusion:**
This equation **does not** represent an inverse proportion.

---

### 3. $$ 0.1x = \frac{5}{y} $$

**Interpretation:**
The equation relates $$ x $$ and $$ y $$ directly:
$$ 0.1x = \frac{5}{y} $$

**Rearrangement:**
Solving for $$ x $$:
$$ x = \frac{5}{0.1y} $$
$$ x = \frac{50}{y} $$

**Inverse Proportion Check:**
Comparing with the general form of an inverse proportion:
$$ x = \frac{k}{y} $$

Here, the constant of proportionality is:
$$ k = 50 $$

**Conclusion:**
This equation **does** represent an inverse proportion with a constant of proportionality $$ k = 50 $$.

---

### Summary

- **Equation 1:** $$ \frac{1}{20} = x = \frac{5}{y} $$  
  **Inverse Proportion:** Yes  
  **Constant of Proportionality (k):** 5

- **Equation 2:** $$ y + \frac{1}{7}x = \frac{1}{2} $$  
  **Inverse Proportion:** No

- **Equation 3:** $$ 0.1x = \frac{5}{y} $$  
  **Inverse Proportion:** Yes  
  **Constant of Proportionality (k):** 50

If you have any further questions or need additional explanations, feel free to ask!
- **Equation 1:** Yes, $$ k = 5 $$ - **Equation 2:** No - **Equation 3:** Yes, $$ k = 50 $$

oftermine whether each equation represents an inverse proportion. It so, find the constant of proportionality. \( \frac{1}{20}=x=\frac{5}{y} \) \( y+\frac{1}{7} x=\frac{1}{2} \) \( 0,1 x=\frac{5}{y} \)

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