Final answer:
The equations of the lines through point P(-5,3) are y = -4x - 17 for the parallel line, and y = 1/4x + 7/4 for the perpendicular line, to the line L: 4x + y = 1.
Step-by-step explanation:
The student asked to write an equation for the line through point P(-5,3) that is (a) parallel to and (b) perpendicular to the line L given by 4x + y = 1. To solve this, we will first identify the slope of line L.
Putting line L in slope-intercept form (y = mx + b) gives us:
y = -4x + 1.
So the slope (m) of line L is -4.
(a) For a line to be parallel to L, it must have the same slope. Thus, the slope of our new line will also be -4. Using the point-slope form of a line (y - y1 = m(x - x1)) with point P(-5,3) gives us the equation:
y - 3 = -4(x + 5),
or
y = -4x - 17.
(b) For a line to be perpendicular to L, its slope must be the negative reciprocal of -4, which is 1/4. Using point-slope form again with point P(-5,3) gives us:
y - 3 = 1/4(x + 5),
or
y = 1/4x + 7/4.