Final answer:
To find a line parallel to 2x + 3y = 4 through (-5, 16), we find the slope of the original line (-2/3) and use it with the given point in the point-slope formula to obtain the new line's equation, which is y = -2/3x + 38/3.
Step-by-step explanation:
To find the equation of a line parallel to 2x + 3y = 4 and passing through the point (-5, 16), first, we need to rewrite the given equation in slope-intercept form to identify the slope. The slope-intercept form is given by y = mx + b, where m is the slope, and b is the y-intercept. After rearranging the given equation, we have:
3y = -2x + 4
y = -2/3x + 4/3
The slope of the original line is -2/3. Since parallel lines have equal slopes, the line we are looking for will also have a slope of -2/3. Now, using the point-slope form of the equation, which is y - y1 = m(x - x1), where (x1, y1) is a known point on the line, we can create the equation of our parallel line:
y - 16 = -2/3(x + 5)
This simplifies to:
y = -2/3x - 10/3 + 48/3
y = -2/3x + 38/3
Therefore, the equation of the line parallel to 2x + 3y = 4 and passing through the point (-5, 16) is y = -2/3x + 38/3.