1. “Teach A Level Maths” Vol. 1: AS Core Modules
1: Straight Lines and Gradients
© Christine Crisp

2. The gradient of the straight line joining the points
and is

3. m is the gradient of the line
The equation of a straight line is
m is the gradient of the line
c is the point where the line meets the y-axis,
the y-intercept
e.g has gradient m =
and y-intercept, c =
gradient = 2 x
intercept on y-axis

4. e.g. Substituting x = 4 in gives
gradient = 2 x
intercept on y-axis
( 4, 7 ) x
The coordinates of any point lying on the line satisfy the equation of the line
e.g. Substituting x = 4 in gives
showing that the point ( 4,7 ) lies on the line.

5. Formula for a Straight Line
Let ( x, y ) be any point on the line x x
Let be a fixed point on the line

6. Finding the equation of a straight line when we know
its gradient, m and
the coordinates of a point on the line (x1,y1).
Using , m is given, so we can find c by
substituting for y1, m and x1.
e.g. Find the equation of the line with gradient
passing through the point
Solution:
(-1, 3) x
So,

7. Using the formula when we are given two points on the line
e.g. Find the equation of the line through the points
Solution: First find the gradient
Now use with
We could use the 2nd point,
(-1, 3) instead of (2, -3)

8. SUMMARY
Equation of a straight line
where m is the gradient and c is the intercept on the y-axis
Gradient of a straight line
where and are points on the line

9. Exercise
1. Find the equation of the line with gradient 2 which passes through the point
Solution:
So,
2. Find the equation of the line through the points
Solution:
So,

10. We sometimes rearrange the equation of a straight line so that zero is on the right-hand side ( r.h.s. )
e.g can be written as
We must take care with the equation in this form.
e.g. Find the gradient of the line with equation
Solution: Rearranging to the form :
so the gradient is

11. Finding the equation of a straight line when we know
its gradient, m and
the coordinates of a point on the line (x1,y1).
Using , m is given, so we can find c by
substituting for y, m and x.
e.g. Find the equation of the line with gradient
passing through the point
Solution:
(-1, 3) x
Add 2 to both sides
C = 5
So,

12. Using the formula when we are given two points on the line
e.g. Find the equation of the line through the points
Solution: First find the gradient
Now use with
We could use the 2nd point,
(-1, 3) instead of (2, -3)
Add 4 to both sides
-3 = -4 + c

14. The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied.
For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

15. If 2 lines have gradients and , then:
They are parallel if
They are perpendicular if
If 2 lines have gradients and , then:
Equation of a straight line
Gradient of a straight line
where and are points on the line
where m is the gradient and c is the intercept on the y-axis
SUMMARY

16. Solution: First find the gradient:
e.g. Find the equation of the line through the points
Now
on the line:
Equation of line is

17. We don’t usually leave fractions ( or decimals ) in equations
We don’t usually leave fractions ( or decimals ) in equations. So, multiplying by 2:
e.g. Find the equation of the line perpendicular to
passing through the point
Solution: The given line has gradient 2. Let
Perpendicular lines:
Equation of a straight line:
on the line