Final answer:
The distance between the parallel lines y = −x + 2 and y = −x + 8 is found to be 3√2 units using the formula for the distance from a point to a line. This method uses perpendicular distance and takes into account the parallel nature of the lines with identical slopes and different y-intercepts.
Step-by-step explanation:
The distance between two parallel lines can be found by measuring a perpendicular distance from any point on one line to the other line. Since the equations of the lines provided are y = −x + 2 and y = −x + 8, we can see that they have the same slope (which is -1) and different y-intercepts. These lines are therefore parallel.
To find the distance between them, we choose an arbitrary point on one of the lines, say (0,2) which is on the first line, and find its distance to the second line using the formula for the distance of a point to a line in the plane. The formula is:
Distance = |Ax_1 + By_1 + C| / √(A^2 + B^2)
Where (x_1, y_1) is the point on the first line, A is the coefficient of x, B is the coefficient of y, and C is the constant term from the second line's equation rearranged to the general form Ax + By + C = 0. For y = −x + 8, this becomes x + y − 8 = 0 (with A=1, B=1, C=−8).
Plugging (0,2) into this formula, the distance is:
Distance = |(1)(0) + (1)(2) - 8| / √(1^2 + 1^2) = |2 - 8| / √2 = 6 / √2 = 6√2 / 2 = 3√2
So the distance between the lines y = −x + 2 and y = −x + 8 is 3√2 units.