Equations of Straight Lines

1. 6 Equations of Straight Lines Case Study 6.1 Equations of Straight Lines 6.2 General Form of Equations of Straight Lines Chapter Summary

2. Do you know how to represent this straight line by an equation? Hm ... Can you tell me? Case Study In the figure, we observe that the straight line passes through the points (0, 0), (1, 1), (2, 2) and (3, 3). For all these points, the x-coordinates and the corresponding y-coordinates are the same. Therefore, we can represent the straight line by the equation yx.

3. 6.1 Equations of Straight Lines A. Introduction In junior forms, we learnt that the graph of a linear equation in two unknowns is a straight line. In the figure, each straight line passes through different points. The line yx passes through the point (2, 2) while the line x 2y 5 passes through the point (1, 2). For every points on a straight line, their x- and y-coordinates must satisfy the equation of the straight line:  For the line x 2y 5 and the point (1, 2), L.H.S.  1  2(2)  5  R.H.S.

4. 6.1 Equations of Straight Lines B. Special Forms of Straight Lines 1. Horizontal Lines (lines that are parallel to the x-axis) In the figure, the straight line L passes through A(–4, 4), B(–1, 4), C(0, 4), D(2, 4) and E(5, 4). The y-coordinates of all these points are equal to 4. The y-intercept is 4. ∴ The equation of L is y 4. In general, the equation of a horizontal line with y-intercept k is given by: y k Notes: The equation of the x-axis is y 0.

5. 6.1 Equations of Straight Lines B. Special Forms of Straight Lines 2. Vertical Lines (lines that are perpendicular to the x-axis) In the figure, the straight line L passes through A(–2, 5), B(–2, 3), C(–2, 0), D(–2, –1) and E(–2, –3). The x-coordinates of all these points are equal to –2. The x-intercept is –2. ∴ The equation of L is x –2. In general, the equation of a horizontal line with x-intercept k is given by: x k Notes: The equation of the y-axis is x 0.

6. 6.1 Equations of Straight Lines This is known as the point-slope form of the equation of the straight line. variables numbers C. Finding Equations of Straight Lines under Different Conditions 1. Given the slope of the straight line and the coordinates of a point on it In the figure, we can draw infinity many lines passing through point A. However, only 1 line can be drawn which passes through point A and is parallel to the Line L. Given a fixed point on a straight line and its slope, it is sufficient to determine that straight line. ConsiderapointA(x1, y1)lyingonastraightlinewith slope m. P(x, y) is any point on the straight line. Slope of PA Slope of the line y – y1 m(x – x1)

7. 6.1 Equations of Straight Lines Consider a straight line L with slope which passes through A(–1, 0). (a) Find the equation of the straight line. (b) Determine whether P(1, 4) and Q(3, 2) lie on the line L. The equation of a straight line can be written in the form of y as the subject. C. Finding Equations of Straight Lines under Different Conditions Example 6.1T Solution: (a) The equation of the straight line:

8. 6.1 Equations of Straight Lines Consider a straight line L with slope which passes through A(–1, 0). (a) Find the equation of the straight line. (b) Determine whether P(1, 4) and Q(3, 2) lie on the line L. (b) Substitute each of the values of into the equation : R.H.S.  R.H.S.  C. Finding Equations of Straight Lines under Different Conditions Example 6.1T Solution: For P(1, 4): L.H.S.  4 For Q(3, 2): L.H.S.  2  2  1 ∵ L.H.S. R.H.S. ∵ L.H.S. R.H.S. ∴P(1, 4) does not lie on the line L. ∴Q(3, 2)lies on the line L.

9. 6.1 Equations of Straight Lines  C. Finding Equations of Straight Lines under Different Conditions Example 6.2T If a line passes through (0, 6) and the inclination is 60, find the equation of the line. Solution: ∵ Inclination  60 ∴ Slope of the line tan 60 The equation of the straight line:

10. 6.1 Equations of Straight Lines This is known as the slope-intercept form of the equation of the straight line. C. Finding Equations of Straight Lines under Different Conditions 2. Given the slope and the y-intercept of the straight line In the figure, line L with slope m passes through a point on the y-axis with the y-intercept c. ∴ Line L passes through the point (0, c). The equation of the straight line: y – y1  m(x – x1) y – c m(x – 0) y – c mx y  mx c

11. 6.1 Equations of Straight Lines Find the equation of the straight line with y-intercept –2 and slope . Since the y-intercept is –2 and the slope of the straight line is given, we can apply the formula directly to find the equation of the straight line. C. Finding Equations of Straight Lines under Different Conditions Example 6.3T Solution: The equation of the straight line:

12. 6.1 Equations of Straight Lines C. Finding Equations of Straight Lines under Different Conditions 3. Given the coordinates of any two points on the straight line In the figure, A(x1, y1) and B(x2, y2) are two points on the straight line L. In this case, we can first find the slope of L by using the formula ∴ y – y1  m(x – x1) Remarks: The above equation looks rather complicated. It is a good practice to identify/find the slope of a straight line first, then follow the point-slope form to find the equation of the straight line.

13. 6.1 Equations of Straight Lines ∴ Slope of the line  C. Finding Equations of Straight Lines under Different Conditions Example 6.4T Find the equation of straight line with x-intercept 2 and y-intercept 6. Solution: The straight line passes through two points (2, 0) and (0, 6).  –3 The equation of the straight line:  Using the point (2, 0):  Using the point (0, 6):

14. (a) Slope of the line   9 6.1 Equations of Straight Lines C. Finding Equations of Straight Lines under Different Conditions Example 6.5T (a) Find the equation of straight line passing through (0, 0) and (–3, –9). (b) If P(3, k) lies on the above straight line, find the value of k. Solution:  3 The equation of the straight line: (b) Substitute x 3, y  k into the equation of straight line, we have k 3(3)

15. 6.2 General Form of Equations of Straight Lines For example: Rewrite into the general form: A. General Form of Equations of Straight Lines From the previous section, we learnt that the equations of straight lines can be written in different ways. In general, the equation of a straight line can be expressed in the form Ax By  C  0, whereA, B and C are constants. This is called the general from of the equation of a straight line. Notes: 1. A, B and C can be any real numbers, but A and B cannot be both zero. 2. The right hand side of the general form of the equation is always zero.

16. 6.2 General Form of Equations of Straight Lines Slope  y-intercept  x-intercept  B. Features of Equations of Straight Lines The general form of the equation of a straight line Ax By  C  0 can be rewritten as: Compare with the slope-intercept form ymxc, we have: If we substitute y 0 into the general form of the equation a straight line Ax By  C  0,

17. 6.2 General Form of Equations of Straight Lines Thus, x  represents a vertical line. Thus, y  represents a horizontal line. The straight line has y-intercept  and the slope  0. 3. Ax By  C  0 can be rewritten as , where B 0. B. Features of Equations of Straight Lines Remarks: 1. If B  0 but A  0, then the equation becomes AxC  0. The straight line does not have any y-intercept and the slope is undefined. 2. If A  0 but B  0, then the equation becomes ByC  0.

18. Slope   2 y-intercept   3 6.2 General Form of Equations of Straight Lines B. Features of Equations of Straight Lines Example 6.6T Consider a straight line L: 6x 3y 9. (a) Find the slope and the y-intercept of the straight line. (b) Find the equation of the straight line with the same y-intercept as L and its slope is 5. Solution: (a) Rewrite the equation of the straight line in the general form, we have 6x 3y 9  0, i.e., 2xy 3  0. (b) The equation of the line: y  5x  3 (or 5x y 3  0)

19. 6.2 General Form of Equations of Straight Lines C. Points of Intersection of Two Lines In junior forms, we learnt the algebraic method of solving simultaneous equations to find the point of intersection of two non-parallel straight lines on the coordinate plane: Consider L1: 2xy 1  0 and L2: y x 5.  Method of substitution  Method of elimination Substituting (2) into (1), we have 2x (x 5)  1  0 (1)  (2), (2xy1) (xy 5)  0 3x 6 x 2 3x 6  0 x 2 Substituting x 2into (2), we have y  2  5  3 Substituting x 2into (2), we have 2  y  5  0 y  3 ∴ The point of intersection of L1 and L2 is (2, 3).

20. 6.2 General Form of Equations of Straight Lines C. Points of Intersection of Two Lines Remarks: Given two straight lines: L1: y m1x c1 L2: y m2x c2 Either one of the following cases will happen: 1. One point of intersection  different slopes, i.e., m1m2 2. No intersection  same slope, i.e., m1m2  different y-intercepts, i.e., c1c2 3. Infinitely many points of intersection  same slope, i.e., m1m2  same y-intercept, i.e., c1c2

21. 6.2 General Form of Equations of Straight Lines C. Points of Intersection of Two Lines Example 6.7T Consider the following straight lines: L1: 4x 3y 7  0 and L2: 3x 2y 18  0 (a) Find the point of intersection of P. (b) Find the equation of the straight line passing through P and the origin. Solution: (a) (1)  3: 12x 9y 21  0 ...... (3) (2)  4: 12x 8y 72  0 ...... (4) (3)  (4): 17y 51  0 y 3 Substituting y 3into (2), we have 3x 2(3) 18  0 x 4 ∴ The point of intersection is (4, 3).

22. 6.2 General Form of Equations of Straight Lines (b) Slope of the line  The straight line passes through (0, 0). Thus the y-intercept is 0.  C. Points of Intersection of Two Lines Example 6.7T Consider the following straight lines: L1: 4x 3y 7  0 and L2: 3x 2y 18  0 (a) Find the point of intersection of P. (b) Find the equation of the straight line passing through P and the origin. Solution: Equation of the line: or 3x 4y  0

23. 6.2 General Form of Equations of Straight Lines D. Parallel Lines and Perpendicular Lines Consider two straight lines L1 and L2 with slopes m1 and m2 respectively. 1. If L1 // L2, then m1m2. Conversely, if m1m2, then L1 // L2. 2. If L1L2, then m1m2 1. Conversely, if m1m2 1, then L1L2.

24. 6.2 General Form of Equations of Straight Lines Slope of L1   Let the equation of L2 be . Substitute (1, 1) into the equation, we obtain c . ∴ D. Parallel Lines and Perpendicular Lines Example 6.8T Consider a straight line L1: 6x 4y 3  0. If another line L2 passes through (1, 1) and is parallel to L1, find the equation of L2. Solution: Alternative Solution: Rewrite the equation of L1: 6x 4y 3  0 The equation of L2: ∵ L1 // L2

25. 6.2 General Form of Equations of Straight Lines D. Parallel Lines and Perpendicular Lines Example 6.9T Consider the following lines L1: 2xy 7  0 and L2: x 4y 1  0. (a) Find the point of intersection of the two lines. (b) Find the equation of the straight line passing through the point of intersection of L1 and L2 which is perpendicular to L2. Solution: (a) (1): 2x y 7  0 (2)  2: 2x 8y 2  0 ...... (3) (1)  (3): 9y 9 0 y 1 Substituting y 1into (2), we have x 4(1) 1  0 x  3 ∴ The point of intersection is (3, 1).

26. 6.2 General Form of Equations of Straight Lines (b) Slope of L2 The required line passes through the point of intersection of L1 and L2, which is (3, 1). slope of the required line   1 D. Parallel Lines and Perpendicular Lines Example 6.9T Consider the following lines L1: 2xy 7  0 and L2: x 4y 1  0. (a) Find the point of intersection of the two lines. (b) Find the equation of the straight line passing through the point of intersection of L1 and L2 which is perpendicular to L2. Solution: Since the required line  L2 , we have slope of the required line  4 Equation of the required line:

27. (c) Given the coordinates of any two points (x1, y1) and (x2, y2) on the straight line: Slope of the line  Then use the point-slope form to find the equation of the straight line. Chapter Summary 6.1 Equations of Straight Lines 1. Special Forms of Straight Lines (a) The equation of a horizontal line with y-intercept k is yk. (b) The equation of a vertical line with x-intercept k is xk. 2. Finding equations of straight lines under different conditions (a) Given the slope of the straight line and the coordinates of point on it: y – y1 m(x – x1) (b) Given the slope and the y-intercept of the straight line: y  mx c

28. (a) slope  (b) x-intercept  (c) y-intercept  Chapter Summary 6.2 General Form of Equations of Straight Lines 1. The general form of the equation of a straight line is AxByC 0. 2. From the general form of the equation of a straight line, for A, B 0, we have: 3. By solving simultaneous equations, we can find the point of intersection of two non-parallel straight lines in the coordinate plane.

29. Chapter Summary 6.2 General Form of Equations of Straight Lines 4. (a) Two straight lines will intersect at one point only if they have different slopes. (b) Two straight lines will not intersect if they have the same slope but different y-intercepts. (c) Two straight lines have infinitely many points of intersection if they have the same slope and y-intercept.