1. 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)

2. Starter:
Write the equation of the line y = ½ x + 5 in the form ax + by + c = 0.
A line, parallel to the line 6x + 3y – 2 = 0, passes through the point (0,3). Work out the equation of the line.

3. Finding the equation of a line
Suppose a line passes through A(x1, y1) with gradient m.
Let P(x, y) be any other point on the line. x y
A(x1, y1)
P(x, y)
y – y1
x – x1 So
This can be rearranged to give y – y1 = m(x – x1).
In general:
This result should be memorized.
The equation of a line through A(x1, y1) with gradient m is
y – y1 = m(x – x1)

4. 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? x y
B(5, 4)
A(3, –2)
The gradient of AB, m =
The equation of this line could also be found by using the given points to find the gradient of the line and using the equation y – y1 = m(x – x1). The gradient could also be substituted into the equation y = mx + c. The value of c can then be found by substituting the values of x and y given by one of the points on the line.
Therefore the gradient of AB is 3 .

5. 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.

6. 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 = ½ x + ½
y = ½ x - 3 ½
2y – x + 7 = 0

7. Task 1 : Work out the gradient of the line joining these pairs of points
(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.

8. Task 2 : Find the equation of the line when:
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)

9. Task 3
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.
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.
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.