C1: The Equation of a Straight
Line, lesson 2
Learning Objective : to be able to
find the equation of a straight line
in the form y – y1 = m(x - x1)
Starter:
1. Write the equation of the line y = ½ x + 5
in the form ax + by + c = 0.
2. A line, parallel to the line 6x + 3y – 2 = 0,
passes through the point (0,3). Work out
the equation of the line.
Finding the equation of a line
Suppose a line passes through A(x1, y1) with gradient m.
y
P(x, y)
A(x1, y1)
y – y1
x – x1
0
x
Let P(x, y) be any other point on the
line.
y  y1
The gradient of AP =
x  x1
y  y1
So
=m
x  x1
This can be rearranged to give y – y1 = m(x – x1).
In general:
The equation of a line through A(x1, y1) with gradient m is
y – y1 = m(x – x1)
Finding the equation of a line
Finding the equation of a line given two points on the
line
A line passes through the points A(3, –2) and
B(5, 4). What is the equation of the line?
The gradient of AB, m = 4  ( 2)
53
y
B(5, 4)
0
x
A(3, –2)
Therefore the gradient of AB is 3 .
Finding the equation of a line
Substituting m = 3 and a co-ordinate pair, A(3, –2),
into the equation
y – y1 = m(x – x1)
gives
y + 2 = 3(x – 3)
y + 2 = 3x – 9
y = 3x – 11
So, the equation of the line passing through the points A(3, –2)
and B(5, 4) is y = 3x – 11.
Example
Find the equation of the line with gradient ½ that
passes through (-1, -4).
Using
y – y1 = m (x – x1)
y- (-4) = ½ ( x – (-1))
y+4=½x+½
y=½x-3½
2y – x + 7 = 0
Task 1 : Work out the gradient of
the line joining these pairs of points
1.
2.
3.
4.
5.
6.
(4, 2) (6, 3)
(-4, 5) (1, 2)
(-3, 4) (7, -6)
(1/4, ½)
(1/2, 2/3)
(3b, -2b) (7b, 2b)
The line joining (3, -5) to (6, a) has a
gradient of 4. Calculate the value of a.
Task 2 : Find the equation of the
line when:
1.
2.
3.
4.
5.
6.
7.
8.
m = 2,
(x1, y1) = ( 2, 5)
m = -1,
(x1, y1) = ( 3, -6)
m = ½,
(x1, y1) = ( -4, 10)
(x1, y1) = ( 2, 4),
(x2, y2) = ( 3, 8)
(x1, y1) = ( 0, 2),
(x2, y2) = ( 3, 5)
(x1, y1) = ( 5, -3),
(x2, y2) = ( 7, 5)
(x1, y1) = ( -1, -5),
(x2, y2) = ( -3, 3)
(x1, y1) = ( 1/3, 2/5), (x2, y2) = ( 2/3, 4/5)
Task 3
1.
The line y = 4x -8 meets the x-axis at the point A. Find
the equation of the line with gradient 3 that passes
through the point A.
2.
The line y = ½ x + 6 meets the x-axis at point B. Find
the equation of the line with gradient 2/3 that passes
through the point B. Write your answer in the form ax +
by + c = 0 where a, b and c are integers.
3.
The lines y = x - 5 and y = 3x – 13 intersect at the point
C. The point D has co-ordinates (-4, 2). Find the
equation of the line that passes through the points C
and D.