Writing Equations in Point Slope Form

Equations of Linear Relations Lesson #3: Writing Equations Using y - y₁ = m(x - x₁) Review In the last lesson, we learned how to write the equation of a straight line using slope-intercept form, namely y = mx + b where m is the slope and b is the y-intercept. There is another equally important method that exists in writing the equation of straight lines. The investigation below explores this method. Investigation Point-Slope Form The graph of y + 3 = 2(x - 1) is shown on the grid. a) Determine the slope of the graph of y + 3 = 2(x 1 - 1). m = 2   b) List the coordinates of point A on the line. (1, -3) c) Compare your answers in a) and b) with the numbers in the equation. y + 3 = 2(x - 1) Slope y-coordinate A The graph of y - 1 = -(2/3)(x + 4) is shown on the grid. d) Determine the slope of the graph of y - 1 = -(2/3)(x + 4). m = -2/3 e) List the coordinates of point B on the line. (-4, 1) f) Compare your answers in d) and e) with the numbers in the equation. y - 1 = -(2/3)(x + 4) Slope y-coordinate B g) Consider the graph of the linear equation y - y₁ = m(x - x₁). Based on your observations in e) and f), state the slope of the line, and write the coordinates of one point on the line. slope m, and the coordinates (x₁, y₁) are the coordinates of one point on the line. Equations of Linear Relations Lesson #3: Writing Equations using y - y₁ = m(x - x₁) Equation of a Line Given the Slope of the Line and a Point on the Line Consider the line with slope 2 passing through the point A(1, -3). The line is shown on the grid. Our objective is to determine the equation of the line. In other words, to find a relation between x and y which is satisfied by every point (x, y) on the line. Let P(x, y) be any point on the line except A. Using the slope formula we have   (y - (-3))/(x - 1) = 2 Cross multiply and solve for y to determine the equation of the line in the form y = mx + b. y + 3 = 2(x - 1) y + 3 = 2x - 2 y = 2x - 5 At this point in the exploration, the equation y = 2x - 5 is valid for all points on the line except A. Note that the coordinates of A, (1, -3), also satisfy the equation, and therefore it is the equation of all points on the line. In the next section, we will use the same procedure to develop a formula   for the equation of any line, given the slope of the line and   a point on the line. The Equation of the Line with slope m through the point (x₁, y₁) Consider the line with slope m passing through the point with coordinates (x₁, y₁). We will use the same procedure as above to show that the equation of the line can be expressed in the form y - y₁ = m(x - x₁). Let P(x, y) be any point on the line distinct from A. Using the slope formula we have m = (y - y₁) / (x - x₁) m(x - x₁) = y - y₁ y - y₁ = m(x - x₁) Equations of Linear Relations Lesson #3: Writing Equations using y - y₁ = m(x - x₁) Point-Slope Equation of a Line: y - y₁ = m(x - x₁) * The point-slope form of the equation of a line is y - y₁ = m(x - x₁), where m is the slope of the line, and (x₁, y₁) represents a point on the line. * To determine the equation of a line in future math courses, the point-slope equation, y - y₁ = m(x - x₁), is used more frequently than the slope-intercept equation, y = mx + b. Note: The point-slope equation is used when we have the slope of a line and the coordinates of any point on that line. It is customary to give the final equation in slope-intercept   form (y = mx + b) or in the general form (Ax + By + C = 0) (to be taught in the next lesson). State the equation, in point-slope form, of the line through the given point and with the given slope. a) (6,   5), m = 3 y - 5 = 3(x - 6) b) (1, -4), m = -4 y + 1 = -4(x - 1) c) (-8, 2), m = 1/2 y - 2 = 1/2(x + 8) In each case, the slope of a line and a point on the line are given. Determine the equation of the line in slope-intercept form, y = mx + b. a) m = 5, L(-5, 2) y - 2 = 5(x + 5) y - 2 = 5x + 25 y = 5x + 27 b) m = -7, point (-3, 4) y - 4 = -7(x + 3) y - 4 = -7x - 21 y = -7x - 17 John and Nicki were solving the following quiz question: "Determine the equation, in slope-intercept form, of a line with slope -2 passing through the point (3, -5)" John could only remember the slope-intercept form y = mx + b, but Nicki remembered the point-slope form y - y₁ = m(x - x₁). Complete their work started below. John's work: y = mx + b y = -2x + b -5 = -2(3) + b -5 = -6 + b b = 1 y = -2x + 1 Nicki's work: y - y₁ = m(x - x₁) y - (-5) = -2(x - 3) y + 5 = -2x + 6 y = -2x + 1 Equations of Linear Relations Lesson #3: Writing Equations using y - y₁ = m(x - x₁) Class Ex. #4 The line on the grid passes through at least two points with integer coordinates. Determine the equation of the line in slope-intercept   form. [Image of a line on a coordinate plane passing through points (-6, 0) and (0, -9)] Use (-6, 0) y - y₁ = m(x - x₁) y - 0 = -3/2(x + 6) y = -3/2x - 9 Class Ex. #5 In each case, state the slope of the line and write the coordinates of a point on the line. a) y + 1 = 1/7(x - 4) m = 1/7, point: (4, -11) b) y - 9 = -5/3(x - 7) m = -5/3, point: (7, 9) c) y = 3(x - 6) m = 3, point: (6, 0) d) y = -3x - 6 m = -3, point: (0, -6) Complete Assignment Questions #1 - #9 Assignment 1. State the equation, in point-slope form, of the line through the given point and with the given slope. a) (9, 3),   4 b) (8, -2), -3 c) (-5,   7), 1 d) (0, 3), 1/2 e) (7, 0), 4 f) (-1/2, -5/4), 5 2. Write the following equations in slope-intercept form y = mx + b. a) y + 1 = 8(x - 2) b) y - 3 = -2(x - 7) c) y - 9 = -11(x + 3)