Lesson 1.3: Equations of Lines
A line on the x-y axes is an infinite collection of points that all satisfy an equation that describes the
relationship between the x-coordinate and the corresponding y-coordinate. The equation of a line can
be written in
form y  mx  b ,
form Ax  By  C , or
form y  y1  mx  x1  . The slope (or gradient) is the ratio between the
vertical displacement and the horizontal displacement, so m 
rise y y 2  y1


. In standard form,
run x x2  x1
A , B ,and C should be co-prime integers (whenever possible), and the value of A should be positive.
e.g. a) Write an equation for the line containing the points 4,3 and 2,1 in slope-intercept
form, standard form, and point-slope form.
We can test whether any particular point is on a line or not by substituting its coordinates into the
equation of the line to establish whether the point satisfies the linear relationship or not.
e.g. b) Determine whether  1,2 and  3,1 are points on the line 2 x  5 y  1 .
The slope of a horizontal line is
, meaning that its equation has the form
. A good
interpretation of the equation of a horizontal line is that the y-coordinate is fixed while the x-coordinate
can be anything. The slope of a vertical line is
(so it cannot be written in slope-intercept
or point-slope form), and has the form
. A good interpretation of the equation of a vertical line
is that the x-coordinate is fixed while the y-coordinate can be anything.
e.g. c) Find the equations of the horizontal and vertical lines passing through the point 4,7 .
Parallel lines have slopes that are
multiply to give a product of
, while perpendicular lines have slopes that are
. Another interpretation of perpendicular lines is that their slopes
.
e.g. d) Write an equation in slope-intercept form for a line that has an x-intercept of  5 and is
parallel to another line with equation 4 x  3 y  2 .
e.g. e) Write an equation in standard form for a line that passes through the point 1,6 and
is perpendicular to another line with equation y  5 
2
x  1 .
7
e.g. f) The graph of 3 x  y  4 is perpendicular to the graph of Ax  2 y  8 . Find A .
Lesson 1.3 (continued)
e.g. g) After driving for 3 hours, Bob is 135 miles from Toronto. After driving for 5 hours, he is
225 miles from Toronto. Write a linear equation in point-slope form that describes this
situation, and state what the slope of the linear equation represents.
A line that rises as you move from left to right has a
slope, while a line that falls as you
move from left to right has a
slope. Furthermore, if the absolute-value of the slope
is large (i.e. significantly bigger than one), then the line is
, while if the absolute-value of the
slope is small (i.e. a fraction close to zero), then the line is
e.g. h) Graph the line with equation 3 x  2 y  4 onto the x-y axes below.
e.g. i) Find the equation of the line graphed below in standard form.