Parallel Lines
Given the equation of a line as:
y = mx + c
The variable m represents the slope or the gradient of the line, and c represents the y-intercept of the line.
If two lines are parallel, they have the same gradient, i.e., their value of m is equal for both and c is different. Note if c was equal also, both lines would be exactly the same, not parallel.
We are given a number of pairs of lines and we must identify which pairs are parallel lines.
a) y = 3x + 4, y = 3x - 8
Comparing the values of m (the coefficient of x) in both equations, we conclude they are parallel lines since m=3 for both and c is different.
b) y = 2x - 17 , y - 2x = 17
We must express the second equation with the y isolated at the left side:
y = 2x + 17
Both lines have the same value of m=2 and different values of c, so they are parallel.
c) y = x - 4 , y = -2x - 4
The first gradient is m=1 and the second gradient is m=-2, thus these lines are not parallel
d) 3y + 3x = 9 , y = x + 3/2
Divide the first equation by 3:
y + x = 3
And solve for y:
y = -x +3
The gradient of this line is -1 and the gradient of the other line is m=1, thus these lines are not parallel.
e) 2x = y + 8 , 2y = x + 8
Solving for y both equations:
y = 2x - 8
y = x/2 + 4
The gradients are m=2 and m=1/2 respectively, thus these lines are not parallel.
f) 3 - x = 3y , y - 3x = 5
Solving for y:
y = -x/3 + 1
y = 3x + 5
The gradients are, respectively m=-1/3 and m=3, thus these lines are not parallel.
g) 4y + 8x = 0 , y = 22 - 2x
Solving for y and rearranging:
y = -2x , y = -2x + 22
The gradients are m=-2 for both lines and the values of c are different, thus these lines are parallel.
h) y = 2x - 8 , 8y + 8x = 9
Solving the second equation for y:
y = -x + 9/8
The gradients are, respectively m=2 and m=-1, thus these lines are not parallel.
The following image summarizes the results by circling the pairs of parallel lines as required.
