To determine which line is parallel to the given graph, we need to find the slope of the line in the graph and then compare it with the slopes of the given equations.

Step-by-Step Solution:

1. Identify the slope of the given graph: The graph shows a line passing through two points:

   • Point 1: (0, -1)
   • Point 2: (3, 1)

   The slope ($m$) of a line passing through points $(x_1, y_1)$ and $(x_2, y_2)$ is given by: $m = \frac{y_2 - y_1}{x_2 - x_1}$ Substituting the coordinates: $m = \frac{1 - (-1)}{3 - 0} = \frac{2}{3}$

2. Rewrite each given equation in slope-intercept form $y = mx + b$ to identify the slopes:

   • Option 1: $3x + 4y = -4$ $4y = -3x - 4 \implies y = -\frac{3}{4}x - 1$ Slope: $m = -\frac{3}{4}$

   • Option 2: $3x - 4y = -12$ $-4y = -3x - 12 \implies y = \frac{3}{4}x + 3$ Slope: $m = \frac{3}{4}$

   • Option 3: $4x + 3y = -6$ $3y = -4x - 6 \implies y = -\frac{4}{3}x - 2$ Slope: $m = -\frac{4}{3}$

   • Option 4: $4x - 3y = 15$ $-3y = -4x + 15 \implies y = \frac{4}{3}x - 5$ Slope: $m = \frac{4}{3}$

3. Compare the slopes: The slope of the given graph is $\frac{2}{3}$. None of the given equations have a slope of $\frac{2}{3}$, hence none of them are parallel to the given graph.

However, based on standard mathematical principles for identifying parallel lines, it seems there might be an error in the graph or equations provided, as none align with the expected parallel slope. Could you confirm the graph details or the equations, or would you like to recheck the points used?

If you have any questions or need further details, please let me know.

Relative Questions:

1. How do you determine the slope of a line from a graph?
2. What is the slope-intercept form of a linear equation?
3. How can you determine if two lines are parallel by comparing their slopes?
4. What are the steps to convert a standard form equation to slope-intercept form?
5. Can you provide an example of two parallel lines in slope-intercept form?

Tip:

When determining the slope of a line from a graph, always pick points that clearly intersect the grid lines to minimize errors in calculation.