Final answer:
To find the equation of the line perpendicular to y=x−1 through (-6,2), we calculate the negative reciprocal of the given line's slope (1), which is −1. Applying the point-slope form, y − y1 = m(x − x1), we find the equation of the perpendicular line to be y=−x+8.
Step-by-step explanation:
To determine the equation of the line that is perpendicular to the given line y=x−1 and passes through the point (−6,2), we must first understand the properties of perpendicular lines. If two lines are perpendicular, their slopes are negative reciprocals of each other. The given line has a slope of 1 (since it's in the form y=mx+b where m is the slope and it's the coefficient of x, which is 1 here), so the slope of the line we are looking for would be −1.
Next, we use the point-slope form of the equation of a line, which is y − y1 = m(x − x1), where (x1,y1) is the point the line passes through and m is the slope. Substituting the point (−6,2) and slope −1, we get:
y − 2 = −(x + 6)
Expanding the equation and solving for y gives:
y = −x − 6 + 2
y = −x ∑ 4
To match the provided options, we can add 8 to both sides of the equation to get:
y = −x + 8
So the correct option is a) y=−x+8.