For each equation, determine whether $$ x $$ and $$ y $$ are directly proportional (that is, if the equation shows direct variation). If so, then find the constant of proportionality (the constant of variation). \begin{tabular}{|l|l}\hline $$ \begin{array}{ll} ext { (a) } y=\frac{8}{5} x & ext { (b) } y=-x \ ext { OProportional } \ ext { Constant of proportionality: } k=\square & ext { OProportional } \ ext { ONot proportional }\end{array} $$\end{tabular} $$ \begin{array}{l} ext { ONot proportional of proportionality: } k=\square\end{array} $$

Question

For each equation, determine whether $$ x $$ and $$ y $$ are directly proportional (that is, if the equation shows direct variation). If so, then find the constant of proportionality (the constant of variation). \begin{tabular}{|l|l}\hline $$ \begin{array}{ll}	ext { (a) } y=\frac{8}{5} x & 	ext { (b) } y=-x \ 	ext { OProportional } \ 	ext { Constant of proportionality: } k=\square & 	ext { OProportional } \ 	ext { ONot proportional }\end{array} $$\end{tabular} $$ \begin{array}{l}	ext { ONot proportional of proportionality: } k=\square\end{array} $$

UpStudy AI · Accepted Answer

Let's analyze each equation to determine whether $$ x $$ and $$ y $$ are directly proportional. If they are, we'll identify the constant of proportionality $$ k $$.

### Direct Proportionality

Two variables $$ x $$ and $$ y $$ are **directly proportional** if their relationship can be expressed in the form:
$$
y = kx
$$
where $$ k $$ is the **constant of proportionality**.

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### (a) $$ y = \frac{8}{5} x $$

- **Form Analysis:** The equation is already in the form $$ y = kx $$, where $$ k = \frac{8}{5} $$.
  
- **Conclusion:**
  - **Directly Proportional:** Yes
  - **Constant of Proportionality:** $$ k = \frac{8}{5} $$

---

### (b) $$ y = -x $$

- **Form Analysis:** The equation can be rewritten as $$ y = (-1)x $$, where $$ k = -1 $$.
  
- **Conclusion:**
  - **Directly Proportional:** Yes
  - **Constant of Proportionality:** $$ k = -1 $$

---

### Summary

| Equation        | Directly Proportional | Constant of Proportionality ($$ k $$) |
|-----------------|-----------------------|----------------------------------------|
| $$ y = \frac{8}{5} x $$ | Yes                   | $$ \frac{8}{5} $$                        |
| $$ y = -x $$            | Yes                   | $$ -1 $$                                |

Both equations (a) and (b) show a direct proportionality between $$ x $$ and $$ y $$ with their respective constants of proportionality.
- **(a) $$ y = \frac{8}{5} x $$:** Directly proportional with $$ k = \frac{8}{5} $$. - **(b) $$ y = -x $$:** Directly proportional with $$ k = -1 $$.

For each equation, determine whether and are directly proportional (that is, if the equation shows direct variation).
If so, then find the constant of proportionality (the constant of variation).

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For each equation, determine whether and are directly proportional (that is, if the equation shows direct variation). If so, then find the constant of proportionality (the constant of variation).