We are tasked with determining which of the given linear equations represents a line that passes through the point (-3, 3) and is parallel to the line shown in the graph.

Step 1: Analyze the slope of the given line in the graph.

The given line in the graph passes through the points (-5, 4) and (0, -1). We can calculate the slope $m$ using the slope formula:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

For points (-5, 4) and (0, -1):

$m = \frac{-1 - 4}{0 - (-5)} = \frac{-5}{5} = -1$

So, the slope of the line in the graph is $-1$.

Step 2: Identify the slope of a parallel line.

For a line to be parallel, it must have the same slope. Therefore, we are looking for a line with slope $-1$ that passes through the point (-3, 3).

Step 3: Test each equation.

Now, let's look at the options:

A. $y = \frac{3}{2}x + 1$ → slope is $\frac{3}{2}$ (Not parallel) B. $y = \frac{3}{2}x - 1$ → slope is $\frac{3}{2}$ (Not parallel) C. $y = -\frac{2}{3}x - 1$ → slope is $-\frac{2}{3}$ (Not parallel) D. $y = -x + 1$ → slope is $-1$ (Parallel)

So, option D has the correct slope.

Step 4: Verify that the line passes through (-3, 3).

Let's check if the point (-3, 3) satisfies the equation $y = -x + 1$.

Substitute $x = -3$ into the equation:

$y = -(-3) + 1 = 3 + 1 = 4$

This does not match the y-coordinate of the point (-3, 3), so option D is incorrect.

Step 5: Equation of the correct line.

We know the slope is $-1$ and the line must pass through (-3, 3). Using the point-slope form of the equation:

$y - y_1 = m(x - x_1)$ where $m = -1$ and $(x_1, y_1) = (-3, 3)$:

$y - 3 = -1(x + 3)$

Simplifying:

$y - 3 = -x - 3$ $y = -x$

Thus, the equation of the line is $y = -x$, which is not directly listed among the options.

The correct answer is not given explicitly in the options. Would you like further clarification or a different approach?

Here are some related questions:

1. How do you determine the slope of a line graphically?
2. What is the point-slope form of a linear equation?
3. How do parallel lines behave in terms of their slopes?
4. What are the steps to find the equation of a line passing through a given point?
5. How can you convert the point-slope form to slope-intercept form?

Tip: When solving for the equation of a line, always remember that parallel lines have identical slopes.