Final answer:
The correct equation of the line passing through point (4, 7) and parallel to the line 4y = 3x + 6 is y = (3/4)x + 4, which is option b).
Step-by-step explanation:
To find the equation of a line that passes through the point (4, 7) and is parallel to the line represented by 4y = 3x + 6, we first need to establish the slope of the given line. By converting the given equation into slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept, we get y = (3/4)x + 3/2. Since parallel lines have the same slope, our new line will also have the slope of (3/4).
Using the slope-point form of a line equation (y - y1 = m(x - x1)), where (x1, y1) is a known point on the line, we substitute the known point (4, 7) and the determined slope (3/4) to find the equation of the parallel line:
y - 7 = (3/4)(x - 4)
Distributing the slope we get:
y - 7 = (3/4)x - 3
Adding 7 to both sides to solve for y, we get:
y = (3/4)x + 4
Hence, the correct equation of the line passing through the point (4, 7) and parallel to the given line 4y = 3x + 6 is y = (3/4)x + 4, which matches option b).