Final answer:
The equation of a line parallel to 5x + y = 13 that passes through the point (15, -31) has the same slope as the given line, which is -5. Using the point-slope form, the equation of the line is found to be y = -5x + 44.
Step-by-step explanation:
The equation of a line that is parallel to another can be found by ensuring that the two lines have the same slope. To find an equation of a line that is parallel to 5x + y = 13 and passes through the point (15, -31), firstly, we need to write the given equation in slope-intercept form to identify the slope. Starting with 5x + y = 13, we can subtract 5x from both sides to get y = -5x + 13, so the slope (m) is -5.
Since parallel lines have equal slopes, the new line will also have a slope of -5. Now, using the point-slope form of a line's equation, which is y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope, we can plug in the known values. Substituting -31 for y1, 15 for x1, and -5 for m gives us y - (-31) = -5(x - 15). Simplifying this, we find y + 31 = -5x + 75.
To put it into slope-intercept form, we subtract 31 from both sides to find the final equation y = -5x + 44. Therefore, the equation of the line parallel to 5x + y = 13 and passing through the point (15, -31) is y = -5x + 44.