Final answer:
Four lines are given and all have the same slope, making them parallel to each other. However, only the lines with different constant terms than the original line are considered distinct parallel lines; these are options 1), 2), and 4). Option 3) represents the same line as the given equation.
Step-by-step explanation:
To determine which lines would be parallel to the line given by the equation -2x + 3y = 12, we need to compare the slopes of the potential lines. The standard form of a line is Ax + By = C, where A/B is the negative of the slope if By is on the left side of the equation. For a line to be parallel to another, their slopes must be the same. In this case, each of the choices provided (-2x + 3y = 6, -2x + 3y = 9, -2x + 3y = 12, -2x + 3y = 15) has the same slope, because the coefficients on x and y are the same.
Therefore, all four equations have lines that are parallel to each other since they all have the slope of 2/3 (if you solve for y, the slope-intercept form would be y = (2/3)x + C). However, only the choices that have a different constant term (C) will be parallel lines that do not coincide with the original line. So the parallel lines to -2x + 3y = 12 would be those with different C values, which are 1), 2), and 4).
To summarize, although choice 3) has the same slope as the original line, it is actually the same line, not a parallel line. Therefore, you can disregard this choice as it does not provide a different line that is parallel to the original line.