Point-slope Form of Equations of Straight Lines

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)

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

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)

Two-point Form of Equations of Straight Lines

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

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

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

Slope-intercept Form of Equations of Straight Lines

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

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

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

Special Straight Lines Equations of Special Straight Lines

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

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 =

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

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

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

General Form of Equation of a Straight Line

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.

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

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

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 =

Possible Intersection of Straight Lines

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?

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)

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.

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).

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

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.