Math Question: Parallel and Perpendicular Lines- Pre-Calculus?

Question

Hi I'm in pre-calc this year and have a test tomorrow. I don't know how to do this. Could someone explain it to me with work???

a. Assume that c does not equal d and a and b are not both zero. Show that ax + by = c and ax + by =d are parallel lines. Explain why the restrictions on a, b, c, and d are necessary.

b. Assume that a and b are not both zero. Show that ax + by = c and bx + ay = d are perpendicular lines. Explain why the restrictions on a and b are necessary.

Thank you so much!

Answer 1

Parallel lines have the same slope. The slope is more apparent if you solve for y.

ax + by = c

by = -ax + c

y = (-a/b)x + c

The slope is -a/b. Do the same for the other line. If a and b are both 0, you have no line. 0=c isn't an equation with solutions for x and y. If c = d they're actually the same line.

for part b, lines are perpendicular iff the slopes multiply to -1. Now you finish.

Answer 2

a) restriction 1: c does not equal d
if they are equal lines are identical

retriction 2: a and b are not both zero
and if a = b = 0 then functions dont define lines

if either a or b = 0 but not both then line is either completely horizontal or vertical which doesn't make them non-parallel but does make the value of b float freely wrt d and c

b) Show that ax + by = c and bx + ay = d are perpendicular lines

equivalent functions are
y = -(a/b) x + c/b
y = -(b/a) x + d/a

since b/a is reciprocal of a/b lines are perpendicular

if both a and b = 0, then functions dont define lines

Answer 3

locate the equation of a line passing by using ( -3, 6) and (a million,2) m = (2 - 6) / (a million + 3) = -4/4 = -a million (y - 6) = -a million(x + 3) y - 6 = -x - 3 y = -x + 3 In slope intercept form x + y = 3 In general form locate the equation of a line passing by using ( 5,-2) and parallel to the line y = 4x-7 m = 4 (y + 2) = 4(x - 5) y + 2 = 2x - 20 y = 2x - 22 locate the equation of a line passing by using 2,a million and perpendicular to the line 4x - 2y = 3 -2y = -4x + 3 y = 2x - 3/2 slope = 2 slope of perpendicular line = -a million / 2 = - a million/2 (y - a million) = (-a million/2)(x - 2) y - a million = (-a million/2)x + a million y = (-a million/2)x + 2

Answer 4

to show that lines are parallel, you have to show that the grad of the lines are equal

for ax+by=c, the grad is -a/b
this is how you get,

by=c-ax
y=c/b - (a/b) x

for ax+by=d, the grad is also -a/b (using the same method)

thus the lines are parallel.

a and b cannot be zero and c & d cannot be equal is because if a is zero and c & d are equal, they are the same line which is x=c (or d), if b is zero and c & d are equal, they are the same line (again) which is y=c(or d).

to show that they are perpendicular, the product of the grad has to be 1

ax+by=c, grad is -a/b as shown above.
bx+ay=d, grad is -b/a

thus, the product of these 2 grad is 1

If a or b are zero, you will not get a perpendicular line

Answer 5

a) Convert both equations into the form
y = mx + b

Lines are parallel if their slopes are the same. so in (a) show that the resulting value in front of the "x" term is the same for both.

b) Again convert both lines to the form
y = mx + b

Lines are perpendicular if their slopes are negative reciprocals of each other. So the resulting term in front of one of the "x" terms should be simply the negative reciprocal of the other.