## Prove that two parallel lines have equal slopes
## Questions on parallel lines
__Question 1__
Which of the lines given by the equations
a) y = 2 x - 3 b) 2 x - y = 2 c) - 4 y + 2 x = 0 d) - 4 y + 8 x = 9
are parallel?
Solution to Question 1
We first need to write each line in slope intercept form y = m x + b and find its slope m.
a) y = 2 x - 3, slope = 2
b) 2 x - y = 2, solve for y, y = 2 x - 2, slope = 2
c) - 4 y + 2 x = 0, solve for y, y = (1 / 2) x, slope = 1 / 2
d) - 4 y + 8 x = 9, solve for y, y = 2 x - 9 / 4, slope = 2
The lines with equal slopes are the lines given in parts a) b) and d) and they are therefore parallel.
__Question 2__
Find k so that the lines with equations 6 k x - 3 y = 9 and - 4 x + 5 y = 7 are parallel.
Solution to Question 2
Solve for y and find the slope of each line
6 k x - 3 y = 9 , solve for y, y = 2 k x - 3 , slope = 2 k
- 4 x + 5 y = 7 , solve for y , y = (4 / 5) x + 7 / 5 , slope = 4 / 5
The two lines are parallel and therefore their slopes are equal; hence
2 k = 4 / 5
Solve for k
k = 2 / 5
__Question 3__
Find a so that the line through the points (1 , 2) and (0 , 3) and the line through the points (a , 2) and ( -2 , 7) are parallel.
Solution to Question 3
Find the slope m_{1} through the points (1 , 2) and (0 , 3)
m_{1} = (3 - 2) /(0 - 1) = - 1
Find the slope m_{2} through the points (a , 2) and (-2 , 7)
m_{2} = (7 - 2) / (-2 - a) = 5 / (-2 - a)
The two lines are parallel, hence their slopes are equal
5 / (-2 - a) = - 1
Solve for a
a = 3
__Question 4__
Find the coordinate of point D so that the points A(0 , 2) , B(2,6), C(8 , 8) and D are the vertices of a parallelogram.
Solution to Question 4
Let (a , b) be the coordinates of point D. For the points A, B, C and D to be the vertices of a parallelogram, segment AB must be parallel to segment DC and segment BC must be parallel to segment AD as shown in the figure below.
slope of segment AB = (6 - 2) / (2 - 0) = 2
slope of segment DC = (8 - b) / (8 - a)
for the segments AB and DC to be parallel, their slopes must be equal, hence the equation
2 = (8 - b) / (8 - a)
cross multiply to obtain
8 - b = 16 - 2 a
slope of segment BC = (8 - 6) / (8 - 2) = 1 / 3
slope of segment AD = (b - 2) / (a - 0) = (b - 2) / a
For the segments BC and AD to be parallel, their slopes must be equal
1 / 3 = (b - 2) / a
cross multiply to obtain
a = 3 b - 6
We now solve the system of the two equations in a and b obtained above and given by
8 - b = 16 - 2 a
a = 3 b - 6
Substitute a by 3 b - 6 in the equation 8 - b = 16 - 2 a to obtain
8 - b = 16 - 2 (3 b - 6)
Solve the above equation for b to obtain
b = 4
then use any of the two equations to find a
a = 6
The coordinates of point D are
D(6 , 4)
__Question 5__
Find the equation of a line L_{1} that passes through the point (2 , 1) and is parallel to the line L_{2} through the points (-1 , 2) and (3 , 4).
Solution to Question 5
The slope m_{2} of the line L_{2} through the points (-1 , 2) and (3 , 4) is given by
m_{2} = (4 - 2) / (3 - (-1) ) = 2 / 4 = 1 / 2
Let m_{1} be the slope of the line L_{1}. L_{1} and L_{2} being parallel their slopes are equal. Hence
m_{1} = m_{2} = 1 / 2
The equation of line L_{1} in point (2 , 1) slope m_{1} form is given by
y - 1 = (1 / 2)(x - 2)
which simplifies to
y = (1 / 2) x
__Question 6__
Find the equation of a line L_{1} that passes through the point (-1 , 1) and is parallel to the line L_{2} with equation 2 y + 4 x = 2.
Solution to Question 6
The slope m_{2} of the line L_{2} is found by first writing its equation in slope intercept form y = m x + b. Hence solve 2 y + 4 x = 2 for y to obtain
y = - 2 x + 1
The slope of L_{2} is
m_{2} = - 2
Since L_{1} and L_{2} are parallel their slopes are equal. Hence
m_{1} = m_{2} = - 2
The equation of line L_{1} in point (-1 , 1) slope m_{1} form is given by
y - 1 = - 2(x - (-1))
which simplifies to
y = - 2 x - 1
## More References and LinksEquations of Lines in Different Forms,
Tutorial on Equation of Line
Line Problems
Equations of Line Through Two Points And Parallel and Perpendicular. |