To determine which equation represents the line shown in the diagram, let's analyze the graph step-by-step.

Step 1: Determine the Slope

The slope ($m$) of a line can be determined by finding the change in $y$ over the change in $x$ ($\Delta y / \Delta x$).

The line intersects the graph at points $(-3, 0)$ and $(0, 3)$:

• From $(-3, 0)$ to $(0, 3)$:
  • The rise ($\Delta y$) is $3$.
  • The run ($\Delta x$) is $3$.

Thus, the slope of the line is: $m = \frac{\Delta y}{\Delta x} = \frac{3}{3} = 1$

Step 2: Identify the y-Intercept

The $y$-intercept is the point where the line crosses the $y$-axis. The line crosses at $y = 3$.

Step 3: Write the Equation of the Line

The equation of a line in slope-intercept form is: $y = mx + b$ Where:

• $m$ is the slope.
• $b$ is the $y$-intercept.

From the graph:

• The slope $m = 1$.
• The y-intercept $b = 3$.

Thus, the equation of the line is: $y = x + 3$

Step 4: Match with the Given Choices

The choices given are:

1. $y = 2x + 3$
2. $y = 3x + \frac{1}{2}$
3. $y = \frac{1}{2}x + 3$
4. $y = 3x + 2$

None of these match $y = x + 3$, which is unexpected, indicating I may have missed something.