Final answer:
The equation of the line parallel to -x + 6y = 36 and passing through (-6, -6) is y = (1/6)x - 7 in slope-intercept form and -x + 6y = -42 in standard form, utilizing the same slope of 1/6 from the given line.
Step-by-step explanation:
To find the equation of the line parallel to -x + 6y = 36 and passing through (-6, -6), first identify the slope of the given line. By transforming the equation into slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept, we rewrite it as y = (1/6)x + 6. This shows that the slope of the given line is 1/6. Since parallel lines have the same slope, our new line will also have a slope of 1/6.
Using the slope 1/6 and the point (-6, -6), we apply the point-slope form y - y1 = m(x - x1), where (x1, y1) is the point the line passes through. Substituting the values, we get y + 6 = (1/6)(x + 6). Simplifying this equation gives us the slope-intercept form: y = (1/6)x - 7.
To express this in standard form, we want Ax + By = C, with A, B, and C being integers, and A being non-negative. After multiplying through by 6 to clear the fraction and rearranging the terms, we get the standard form as -x + 6y = -42.