Final answer:
To find the equation of a line parallel to 3x+4y=5 and through the point (3,-2), determine the slope of the given line, which is -3/4, then apply the point-slope equation using the point (3,-2) and this slope to get y + 2 = -3/4(x - 3).
Step-by-step explanation:
The question asks us to determine the equation of a line that passes through the point (3,-2) and is parallel to the line represented by the equation 3x+4y=5. Parallel lines have the same slope, so first, we need to find the slope of the given line by rearranging it into slope-intercept form, which is y=mx+b where 'm' represents the slope.
Rearranging 3x + 4y = 5 into slope-intercept form gives us y = -¾ x + ⅔. Thus, the slope of the given line is -¾. Since parallel lines have the same slope, the line we are looking for will also have a slope of -¾.
To find the specific equation of our line, we use the point-slope form of a line, which is y - y1 = m(x - x1), 'm' being the slope and (x1, y1) being the point the line passes through.
Substituting the given point (3,-2) and the slope -¾ into the point-slope form gives us y - (-2) = -¾(x - 3). Simplifying, we get the final equation of the line parallel to 3x+4y=5 that passes through (3,-2) as y + 2 = -¾(x - 3).