Straight Lines
Positive and Negative Gradients
If a line has gradient m = 1 then, providing that the scales
are the same for both axes, it makes an angle of 45Âº with the
positive xaxis. If m > 1 then the gradient is steeper. If 0
< m < 1 then the line makes an angle between 0Âº and 45Âº
with the positive xaxis.
The diagram below illustrates lines similar to those of the
previous diagram except with negative gradients. They are the
mirror images of thestraight lines which are shown in the
previous diagram, with the y axis actingas the mirror.
Exercise 1
In each of the following either the coordinates of twopoints,
P,Q are given, or the coordinates of a single point R and
agradient m . In each case, find the equation of the line.
(a) P (1 , 1) , Q (2 ,  1)
(b) R (1 , 2) , m = 2
(c) P (  1 , 2) , Q (1 ,  3)
(d) R (  2 , 1) , m = 4
Solutions
(a) Let the line be y = mx + c . Since both P
(1 , 1) andQ (2 ,  1) lie on the line, both sets of coordinates
must satisfy theequation. Thus we have
1 = m Ã— 1 + c using the coordinates of P
 1 = m Ã— 2 + c using the coordinates of Q
or m + c = 12 m + c =  1 .
This is a set of simultaneous equations which can be solved to
givem =  2 and c = 3 .
The required equation is thus
y =  2 x + 3 .
Substituting the coordinates for P and then Q into this
equation will confirm that this line passes through both of these
points.
(b) Since m = 2 , the equation must have the
form y =2 x + c and only the value of c remains to be found. The
line passesthrough R (1 , 2) so the coordinates of this point
must satisfy the equation. Thus
y = 2 x + c
2 = 2 Ã— 1 + c using the coordinates of R
giving c = 0 .
The equation of the line is now
y = 2 x .
(c) Let the line be y = mx + c . Since both P
(  1 , 2) and Q (1 ,  3) lie on the line, both sets of
coordinates must satisfy theequation. Thus we have
2 = m Ã— (  1 ) + c using the coordinates of P
 3 = m Ã— 1 + c using the coordinates of Q
or  m + c = 2m + c =  3 .
This is a set of simultaneous equations which can be solved to
givem =  5 / 2 and c =  1 / 2 .
The required equation isthus
y =  52 x  12 .
Substituting the coordinates for P and then Q into this
equation will confirm that this line passes through both of these
points.
(d) Since m = 4 , the equation must have the
form y =4 x + c and only c remains to be found. The line passes
throughR (  2 , 1) so the coordinates of this point must satisfy
the equation. Thus
y = 4 x + c
1 = 4 Ã— (  2) + c using the coordinates of R
or  8 + c = 1c = 9 ,
and the equation of the line is
y = 4 x + 9 .
Example 5
Two lines are described as follows: the first has gradient1
and passes through the point R(2,1) ; the second passes through
thetwo points with coordinates P(2,0) and Q(0,4) . Find the
equation ofboth lines and find the coordinates of their point of
intersection.
Solution
The first line has gradient m =  1 so it must be y =(  1) x
+ c , i.e. y =  x + c , for some c . Since the line passes
throughthe point R (2 , 1) these values of x, y must satisfy the
equation. Thus2 =  (1)+ c so c = 3 . The first line therefore
has equation y =  x +3 . For the second case both points lie on
the line and so satisfy theequation. If the equation is y = mx +
c then putting these values intothe equation gives
y = mx + c
0 = 2 m + c (using the coordinates of P )
4 = c (using the coordinates of Q )
These equations yield m =  2 and c = 4 .
The second line thus has the equation y =  2 x +4 . The
equations of the two lines can now be rewritten as
y + x = 3 (1)
y + 2 x = 4 (2)
which is a pair of simultaneous equations. Subtracting
equation (1) from equation (2) gives x = 1 and substituting this
into the first equationthen yields y = 2 . The point of
intersection thus has coordinates x = 1 , y = 2 . (By
substituting these coordinates into equation (2) and verifying
that they satisfy the equation, it can be checked that this is
also a point on the second line.)
Some Useful Facts
• Parallel lines have the same gradient. Thus, for
example, the lines with equations y = 3 x + 7 andy = 3 x  2 are
parallel.
• Lines parallel to the x axis have equations of the
form y = k, for some constant, k .
• Lines parallel to the y axis (when m = 0 ) have
equations of the form x = k , for some constant, k .
• The larger the absolute value of m , the
‘steeper’ the slope of the line.
• If two lines intersect at right angles then the product
of their gradients is  1 . The lines y =  7 x +4and y = (1 / 7)
x + 5 , for example, intersect each other at right angles.
