Final answer:
The equation of the line perpendicular to y = -3/8x - 56 and goes through the point (12,2) is y = (8/3)x - 30, determined by finding the negative reciprocal of the given line's slope and then using the point-slope form.
Step-by-step explanation:
To find the equation of a line that is perpendicular to a given line, you first need to determine the slope of the given line. The slope of the line y = -3/8x - 56 is -3/8. A perpendicular line will have a slope that is the negative reciprocal of this slope, which is 8/3.
Next, you use the point-slope form of a line's equation, which is y - y1 = m(x - x1), where m is the slope of the line and (x1, y1) is a point on the line. Using the point given (12, 2), and the perpendicular slope 8/3, we get:
y - 2 = (8/3)(x - 12).
To put this equation into slope-intercept form (y = mx + b), expand and simplify:
y - 2 = (8/3)x - (8/3)×12
y - 2 = (8/3)x - 32
Finally, add 2 to both sides to solve for y:
y = (8/3)x - 30
So, the equation of the line perpendicular to y = -3/8x - 56 and passing through the point (12, 2) is y = (8/3)x - 30.