C1: The Equation of a Straight
Line
Learning Objective : to be able to
find the equation of a straight line
and to express it in different forms
Starter:
On axes from -5 to 5, sketch the straight
lines:
y=3
x=2
x = -1
y=x
y=x+2
What is the equation of the x-axis?
The equation of a straight line
The equation of a straight line can be written in several forms.
You are probably most familiar with the equation written in the
form y = mx + c.
The value of m tells us the gradient
y
of the line.
The value of c tells us where the
line cuts the y-axis.
m
1
This is called the y-intercept and it
has the coordinates (0, c).
c
0
x
For example, the line y = 3x + 4 has
a gradient of 3 and crosses the
y-axis at the point (0, 4).
The equation of a straight line
One more way to give the equation of a straight line is in the
form
ax + by + c = 0.
This form is often used when the required equation contains
fractions. For example, the equation
y = 34 x  21
can be rewritten without fractions as
4y – 3x + 2 = 0.
It is important to note that any straight line can be written in
the form ax + by + c = 0.
In particular, equations of the form x = c can be written in the
form ax + by + c = 0 but cannot be written in the form y = mx + c.
Example
Express the line 3x + 5y – 12 = 0 in the form
y = mx + c and hence state the gradient and
intercept of the line.
3x + 5y – 12 = 0
5y = -3x + 12
y = -3/5 x + 12/5
Hence,
gradient = -3/5 or – 0.6, intercept = 12/5 or 2.4
Task 1 : Work out the gradient and
intercept of these lines
1.
2.
3.
4.
5.
6.
7.
8.
y = -2x + 5
y=7–x
y = -2/3 x
2x – 4y + 5 = 0
10x – 5y + 1 = 0
-x + 2y – 4 = 0
7x – 2y + 3 = 0
9x + 6y + 2 = 0
Task 2 : Write these lines in the
form ax + by + c = 0
1.
2.
3.
4.
5.
6.
7.
8.
y = 4x + 3
y = 3x -2
y = -6x +7
y = 4/5 x – 6
y = 7/3 x
y = 2x – 4/7
y = 2/3 x + 5/6
y = 3/5 x + ½
Task 3
1. A line is parallel to the line y = 5x + 8 and its
intercept on the y-axis is (0,3). Write down the
equation of the line.
2. The line y = 6x – 18 meets the x-axis at point
P. Work out the co-ordinates of P.
3. A line is parallel to the the line y = -2/5 x + 1
and its intercept on the y –axis is (0, -4). Work
out the equation of the line. Write your answer
in the form ax + by + c = 0 where a, b and c
are integers.