$$ \left. \begin{array} { | c | } \hline \frac { 1 } { 3 } \ \hline ext { EQUATION OF PROPORTIONALITY } \ \hline y = \frac { 1 } { 3 } x \ \hline \end{array} ight. $$

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$$ \left. \begin{array} { | c | } \hline \frac { 1 } { 3 } \ \hline 	ext { EQUATION OF PROPORTIONALITY } \ \hline y = \frac { 1 } { 3 } x \ \hline \end{array} ight. $$

UpStudy AI · Accepted Answer

The equation you've provided, $$ y = \frac{1}{3} x $$, represents a linear relationship between the variables $$ y $$ and $$ x $$. This equation indicates that $$ y $$ is directly proportional to $$ x $$ with a proportionality constant of $$ \frac{1}{3} $$.

### Explanation of the Equation

1. **Proportionality**: The equation shows that for every unit increase in $$ x $$, $$ y $$ increases by $$ \frac{1}{3} $$ of that unit. This means that $$ y $$ is one-third of $$ x $$.

2. **Graphical Representation**: If you were to graph this equation, it would be a straight line that passes through the origin (0,0) with a slope of $$ \frac{1}{3} $$. This slope indicates that for every 3 units you move horizontally (along the $$ x $$-axis), you move 1 unit vertically (along the $$ y $$-axis).

3. **Intercepts**:
   - **Y-intercept**: The y-intercept occurs when $$ x = 0 $$. Substituting $$ x = 0 $$ into the equation gives $$ y = 0 $$. Thus, the y-intercept is at the point (0,0).
   - **X-intercept**: The x-intercept occurs when $$ y = 0 $$. Setting $$ y = 0 $$ in the equation gives $$ 0 = \frac{1}{3} x $$, which also results in $$ x = 0 $$. Thus, the x-intercept is also at the point (0,0).

### Summary
The equation $$ y = \frac{1}{3} x $$ describes a linear relationship where $$ y $$ is directly proportional to $$ x $$ with a constant of proportionality of $$ \frac{1}{3} $$. The line representing this equation passes through the origin and has a gentle slope.
The equation $$ y = \frac{1}{3} x $$ shows that $$ y $$ is directly proportional to $$ x $$ with a constant of $$ \frac{1}{3} $$. It represents a straight line passing through the origin with a slope of $$ \frac{1}{3} $$.

\( \left. \begin{array} { | c | } \hline \frac { 1 } { 3 } \\ \hline \text { EQUATION OF PROPORTIONALITY } \\ \hline y = \frac { 1 } { 3 } x \\ \hline \end{array} \right. \)

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