Parallel lines are the lines that never intersect.
In the coordinate plane, they would look like as shown below.
If we take a closer look at these two lines, the slope of both the lines is 1/2.
This can be generalized to any pair of parallel lines. Parallel lines always have the same slope and different y-intercepts.
Postulate (Slopes of Parallel Lines) :
In a coordinate plane, two lines are parallel if and only if they have the same slope.
The slope of a non vertical line is the ratio of the vertical change (the rise) to the horizontal change (the run).
If the line passes through the points (x_{1}, y_{1}) and (x_{2}, y_{2}), then the slope is given by
Slope = rise/run
Slope = (y_{2} - y_{1})/(x_{2} - x_{1})
Usually, slope is represented by the variable m.
Example 1 :
In the diagram given below, find the slope of each line. Determine whether the lines j_{1 }and j_{2 }are parallel.
Solution :
Line j_{1} has a slope of
m_{2} = 4/2
= 2
Line j_{2} has a slope of
m_{2} = 2/1
= 2
Since the slope of the lines j_{1 }and j_{2 }are equal, the lines j_{1 }and j_{2 }are parallel.
Example 2 :
In the diagram given below, find the slope of each line. Which lines are parallel?
Solution :
Part 1 :
Find the slope of the line k_{1}. Line k_{1} is passing through the points (0, 6) and (2, 0).
Let (x_{1}, y_{1}) = (0, 6) and (x_{2}, y_{2}) = (2, 0).
Slope (k_{1}) = (y_{2} - y_{1})/(x_{2} - x_{1})
= (0 - 6)/(2 - 0)
= -6/2
= - 3
Part 2 :
Find the slope of the line k_{2}. Line k_{2} is passing through the points (-2, 6) and (0, 1).
Let (x_{1}, y_{1}) = (-2, 6) and (x_{2}, y_{2}) = (0, 1).
Slope (k_{2}) = (y_{2} - y_{1})/(x_{2} - x_{1})
= (1 - 6)/[0 -(-2)]
= (1 - 6)/[0 + 2]
= -5/2
Part 3 :
Find the slope of the line k_{3}. Line k_{3} is passing through the points (-6, 5) and (-4, 0).
Let (x_{1}, y_{1}) = (-6, 5) and (x_{2}, y_{2}) = (-4, 0).
Slope (k_{3}) = (y_{2} - y_{1})/(x_{2} - x_{1})
= (0 - 5)/[-4 - (-6)]
= -5/(-4 + 6)
= -5/2
Compare the slopes. Because k_{2} and k_{3} have the same slope, they are parallel. Line k_{1} has a different slope, so it is not parallel to either of the lines.
Example 3 :
Write an equation of the line through the point (2, 3) that has a slope 5.
Solution :
Slope-intercept form equation of a line :
y = mx + b ----(1)
Substitute (x, y) = (2, 3) and m = 5.
3 = 5(2) + b
Simplify.
3 = 10 + b
Subtract 3 from both sides.
-7 = b
The equation of the required line is
(1)----> y = 5x - 7
Example 4 :
In the diagram given below, line n_{1} has the equation
y = -x/3 -1
Line n_{2} is parallel to the line n_{1} and passes through the point (3, 2).
Write the equation of the line n_{2}.
Solution :
The slope of the line n_{1} is -1/3. Because the lines n_{1} and n_{2} are parallel, they have the same slope. So, the slope of the line n_{2} is also -1/3.
Slope-intercept form equation of a line :
y = mx + b ----(1)
Because the line n_{2} is passing through (3, 2), substitute
(x, y) = (3, 2) and m = -1/3
2 = (-1/3)(3) + b
Simplify.
2 = -1 + b
Add 1 to both sides.
3 = b
The equation of the required line is
(1)----> y = (-1/3)x + 3
y = -x/3 + 3
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