1. Point-slope Form of Equations of Straight Lines

2. How can we find the equation of L?
Suppose we are given a point A(x1, y1) on a straight line L with slope m.
How can we find the equation of L?
Let P(x, y) be a point on L.
Slope of AP
= slope of L
y – y1
i.e. m =
x – x1
y – y1 = m(x – x1)

3. Point-slope Form
The equation of L passing through A(x1, y1) with slope m is given by:
y – y1 = m(x – x1)

4. Follow-up question
Find the equation of the straight line passing through (–1, 6) with slope –2.
The equation of the straight line is
y - 6 = -2[x - (-1)]
y - 6 = -2x - 2
 y = -2x + 4
 y – y1 = m(x – x1)

5. Two-point Form of Equations of Straight Lines

6. Suppose we are given two points on a straight line L.
How can we find the equation of L?
Slope of L = 1 2 x y -
By the point-slope form, we have 1 2 x y - è ç æ y - y = m ) ( 1 x - 1

7. Two-point Form
The equation of L passing through two points A(x1, y1) and B(x2, y2) is given by: ) ( 1 2 x y - = è ç æ

8. Follow-up question
Find the equation of the straight line passing through (3, 0) and (1, 4).
The equation of the straight line is 

9. Slope-intercept Form of Equations of Straight Lines

10. How can we find the equation of L?
Suppose we are given the slope and the y-intercept of a straight line L.
How can we find the equation of L?
By the point-slope form, we have
y - c = m(x - 0)
 L passes through (0, c).
y - c = mx
y = mx + c

11. Slope-intercept Form
The equation of L with slope m and y-intercept c is given by :
y = mx + c

12. Follow-up question
Find the equation of the straight line with y-intercept –2 and slope 3.
The equation of the straight line is
y = 3x + (-2)
 y = 3x - 2
 y = mx + c

13. Special Straight Lines
Equations of
Special Straight Lines

14. Oblique Lines Passing Through the Origin
Case 1: Given the slope m
By the slope-intercept form,
y = mx + 0
 y-intercept = 0
The equation of the straight
line L is:
y = mx

15. Case 2: Given the point (a, b)
Slope of L - = a b a b =
The equation of the straight line L is: x a b y =

16. Follow-up question
Find the equation of the straight line passing through the origin and (2, 6).
Slope - = 2 6 3 =
The equation of the straight line is
y = 3x + 0
 y = mx + 0
∴ y = 3x
y-intercept

17. Horizontal Lines
All the points lying on a horizontal line have the same
(-2, ) b
(-1, ) b
(1, ) b
(2, ) b
y-coordinate
The equation of the horizontal line shown is:
y = b

18. Vertical Lines All the points lying on a vertical line have the same .
( , 2) a
( , 1) a
x-coordinate
( , -1) a
( , 2) a
The equation of the vertical line shown is:
x = a

19. General Form of
Equation of a Straight Line

20. General Form of Equation of a Straight Line
Two unknowns x, y
Ax + By + C = 0
A, B and C are constants.
A and B are NOT both zero.

21. Follow-up question Rewrite the equation of the straight line
L: 2y = -4x + 3 into the general form.
2y = -4x + 3
4x + 2y - 3 = 0
 General form:
Ax + By + C = 0
C = -3
B = 2
A = 4

22. For a straight line L in the general form Ax + By + C = 0, we have:
Substitute y = 0. C By Ax = + C B Ax = +
(0) C Ax By - = C Ax - = A C C y = - x - x = - B B A
Therefore, A C C
slope = - , y -
intercept = -
and x -
intercept = - B B A

23. Follow-up question
Find the slope, the y-intercept and the x-intercept of the straight line 2x + y - 4 = 0.
From the equation, A = 2, B = 1 and C = -4. B A - = 1 2 - =
 Slope 2 - = B C - = ç è æ - = 1 4
y-intercept  4 = A C - = ç è æ - = 2 4
x-intercept  2 =

24. Possible Intersection of Straight Lines

25. Then, how can I find the coordinates of their intersection?
Given two non-parallel lines L1 and L2, they must intersect.
Then, how can I find the coordinates of their intersection?

26. Consider the following two straight lines.
L1: x + y - 4 = 0 ……(1)
L2: x - y - 2 = 0 ……(2)
If (x0, y0) is the intersection of L1 and L2, then L1 L2 O y x
(x0, y0) satisfies both equations of L1 and L2.
(x0, y0) lies on  both L1 and L2.
(x0, y0)

27. Consider the following two straight lines.
L1: x + y - 4 = 0 ……(1)
L2: x - y - 2 = 0 ……(2)
Since the coordinates of the intersection satisfies both equations of L1 and L2,
we can solve to find the
coordinates of the intersection.

28. Consider the following two straight lines.
L1: x + y - 4 = 0 ……(1)
L2: x - y - 2 = 0 ……(2)
(1) + (2): 2x - 6 = 0
x = 3
By substituting x = 3 into (1), we have
3 + y - 4 = 0
y = 1
∴ L1 and L2 intersect at (3, 1).

29. Number of intersections of two straight lines
Case 1
Case 2
Case 3
Condition
unequal slopes
equal slope and
unequal y-intercepts
equal slope and equal y-intercept
No. of intersections
one intersection
no intersections
an infinite no. of intersections

30. Follow-up question
Determine the number of intersections between the two straight lines L1 : 3x + 2y - 5 = 0 and L2 : 6x + 2y - 3 = 0.
Slope of L1 = - 2 3
 For a straight line Ax + By + C = 0,
Slope of L2 = = - 2 6 ∵
Slope of L1  slope of L2
∴ L1 and L2 have one intersection.