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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) Starter: 1. Write the equation of the line y = ½ x + 5 in the form ax + by + c = 0. 2. A line, parallel to the line 6x + 3y – 2 = 0, passes through the point (0,3). Work out the equation of the line. Finding the equation of a line Suppose a line passes through A(x1, y1) with gradient m. y P(x, y) A(x1, y1) y – y1 x – x1 0 x Let P(x, y) be any other point on the line. y y1 The gradient of AP = x x1 y y1 So =m x x1 This can be rearranged to give y – y1 = m(x – x1). In general: The equation of a line through A(x1, y1) with gradient m is y – y1 = m(x – x1) 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? The gradient of AB, m = 4 ( 2) 53 y B(5, 4) 0 x A(3, –2) Therefore the gradient of AB is 3 . 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. 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=½x+½ y=½x-3½ 2y – x + 7 = 0 Task 1 : Work out the gradient of the line joining these pairs of points 1. 2. 3. 4. 5. 6. (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. Task 2 : Find the equation of the line when: 1. 2. 3. 4. 5. 6. 7. 8. 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) Task 3 1. 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. 2. 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. 3. 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.