To find the equation of the line that passes through a given point and is perpendicular to a given line, we first need to find the slope (m) of the new line.
The slope of a line perpendicular to the given line is the negative reciprocal of the given line's slope. The given line has a slope of -1/3, so the slope of the line perpendicular to it is -1 divided by -1/3, which simplifies to 3.
We use the point-slope form of a line, y - y1 = m(x - x1), where (x1, y1) is a point on the line. In this case, x1 is 4 and y1 is 7.
Substituting these values and the newly calculated slope m=3 into the point-slope form gives us:
y - 7 = 3(x - 4)
Now we solve this equation for y to get it into the standard y = mx + b form. Distributing the 3 gives y - 7 = 3x - 12.
Adding 7 to both sides to isolate y on left side gives us: y = 3x - 5.
So, the equation of the line that passes through the point (4,7) and is perpendicular to the line y=-(1)/(3)x+5 is y = 3x - 5. Hence, the value of b (the y-intercept) is -5.