ALGEBRA TWO Section 2.4 - Writing Equations of Lines CHAPTER TWO: LINEAR EQUATIONS AND FUNCTIONS Section 2.4 - Writing Equations of Lines

LEARNING GOALS Goal One - Write linear equations. Goal Two - Write direct variation equations.

Writing an Equation of a Line SLOPE - INTERCEPT FORM: Given the slope m and the y-intercept b, use the equation y = mx + b. POINT - SLOPE FORM: Given the slope m and a point (x1, y1), use the equation y - y1 = m (x - x1). TWO POINTS: Given two points (x1, y1) and (x2, y2), use the formula m = (y2 - y1) / (x2 - x1) to find the slope m. Then use the point-slope form with this slope and either of the given points to write an equation of the line. 1

PROBLEM: Write an equation of the line that has m = -2/3 and b = -2 Writing an Equation Given the Slope and y-intercept PROBLEM: Write an equation of the line that has m = -2/3 and b = -2 SOLUTION y = mx + b Use the slope-intercept form. y = (-2/3)x - 2 Substitute (-2/3) for m and (-2) for b. An equation of the line is y = (-2/3)x - 2

PROBLEM: Write an equation of the line that has m = 3 and b = 0 Writing an Equation Given the Slope and y-intercept PROBLEM: Write an equation of the line that has m = 3 and b = 0 SOLUTION y = mx + b Use the slope-intercept form. y = (3)x + 0 Substitute (3) for m and (0) for b. An equation of the line is y = 3x

PROBLEM: Write an equation of the line that has m = 3/4 and b = 2 Writing an Equation Given the Slope and y-intercept PROBLEM: Write an equation of the line that has m = 3/4 and b = 2 SOLUTION y = mx + b Use the slope-intercept form. y = (3/4)x + 2 Substitute (3/4) for m and (2) for b. An equation of the line is y = (3/4)x + 2

PROBLEM: Write an equation of the line that has m = -2 and b = -3 Writing an Equation Given the Slope and y-intercept PROBLEM: Write an equation of the line that has m = -2 and b = -3 SOLUTION y = mx + b Use the slope-intercept form. y = (-2)x + (-3) Substitute (-2) for m and (-3) for b. An equation of the line is y = -2x - 3

Writing an Equation Given the Slope and a Point PROBLEM: Write an equation of the line that passes through (2, -1) and has a slope of -5. SOLUTION y - y1 = m(x - x1) Use the point-slope form. y - (-1) = (-5)(x - (2)) Substitute (-5) for m and (2) for x1 and (-1) for y1. y + 1 = -5x + 10 Use distributive property y = -5x + 9 Simplify The equation for the line is y = -5x + 9

Writing an Equation Given the Slope and a Point PROBLEM: Write an equation of the line that passes through (-3, -2) and has a slope of (1/3). SOLUTION y - y1 = m(x - x1) Use the point-slope form. y - (-2) = (1/3)(x - (-3)) Substitute (1/3) for m and (-3) for x1 and (-2) for y1. y + 2 = (1/3)x + (1) Use distributive property y = (1/3)x - 1 Simplify The equation for the line is y = (1/3)x - 1

Writing an Equation Given Two Points The slope for the line is m = 7/3 PROBLEM: Write an equation of the line that passes through (-1, -3) and (2, 4). SOLUTION m = (y2 - y1) /(x2 - x1) Write formula for slope. m = (4 - (-3))(2 - (-1)) Substitute. m = (4 + 3) / (2 + 1) Simplify m = 7/3 Simplify The slope for the line is m = 7/3

Writing an Equation Given Two Points PROBLEM: Write an equation of the line that passes through (-1, -3) and (2, 4). SOLUTION y - y1 = m(x - x1) Use the point-slope form. y - (-3) = (7/3)(x - (-1)) Substitute (7/3) for m and (-1) for x1 and (-3) for y1. y + 3 = (7/3)x + 7/3 Use distributive property y = (7/3)x - (2/3) Simplify The equation for the line is y = (7/3)x – (2/3)

Writing an Equation Given Two Points The slope for the line is m = -3 PROBLEM: Write an equation of the line that passes through (2, 5) and (4, -1). SOLUTION m = (y2 - y1) /(x2 - x1) Write formula for slope. m = (-1 - (5))(4 - (2)) Substitute. m = (-6) / (2) Simplify m = -3 Simplify The slope for the line is m = -3

Writing an Equation Given Two Points PROBLEM: Write an equation of the line that passes through (2, 5) and (4, -1). SOLUTION y - y1 = m(x - x1) Use the point-slope form. y - (5) = (-3)(x - (2)) Substitute (-3) for m and (2) for x1 and (5) for y1. y - 5 = (-3)x + (6) Use distributive property y = (-3)x + 11 Simplify The equation for the line is y = -3x + 11

Writing an Equation Given Two Points The slope for the line is m = 1 PROBLEM: Write an equation of the line that passes through (-2, 1) and (4, 7). SOLUTION m = (y2 - y1) /(x2 - x1) Write formula for slope. m = (7 - (1))(4 - (-2)) Substitute. m = (6) / (4 + 2) Simplify m = 6/6 = 1 Simplify The slope for the line is m = 1

Writing an Equation Given Two Points PROBLEM: Write an equation of the line that passes through (-2, 1) and (4, 7). SOLUTION y - y1 = m(x - x1) Use the point-slope form. y - (1) = (1)(x - (-2)) Substitute (1) for m and (-2) for x1 and (1) for y1. y - 1 = x + (2) Use distributive property y = x + 3 Simplify The equation for the line is y = x + 3

VOCABULARY Two variables x and y show direct variation provided y = kx and k =/= 0. The nonzero constant k is called the constant of variation. 1

Writing and Using a Direct Variation Equation PROBLEM: The variables x and y vary directly, and y = 3 when x = -4. Write an equation that relates the variables. Then find y when x = 4. SOLUTION y = kx Write direct variation equation. (3) = k(-4) Substitute (-4) for x and (3) for y. -3/4 = k Solve for k. The value of k = (-3/4). The equation is y = (-3/4)x

Writing and Using a Direct Variation Equation PROBLEM: The variables x and y vary directly, and y = 3 when x = -4. Write an equation that relates the variables. Then find y when x = 4. SOLUTION y = (-3/4)x Write direct variation equation. y = (-3/4)(4) Substitute (4) for x. y = -3 Solve for y. The value of y = -3 when x = 4.

Writing and Using a Direct Variation Equation PROBLEM: The variables x and y vary directly, and x = 10 when y = 100. Write an equation that relates the variables. Then find y when x = - 2. SOLUTION y = kx Write direct variation equation. (100) = k(10) Substitute (10) for x and (100) for y. 10 = k Solve for k. The value of k = 10. The equation is y = 10x

The value of y = -20 when x = -2. Writing and Using a Direct Variation Equation PROBLEM: The variables x and y vary directly, and x = 10 when y = 100. Write an equation that relates the variables. Then find y when x = - 2. SOLUTION y = 10x Write direct variation equation. y = 10(-2) Substitute (-2) for x. y = -20 Solve for y. The value of y = -20 when x = -2.

Writing and Using a Direct Variation Equation PROBLEM: The variables x and y vary directly, and x = -3 when y = 12. Write an equation that relates the variables. Then find y when x = -2. SOLUTION y = kx Write direct variation equation. (12) = k(-3) Substitute (-3) for x and (12) for y. -4 = k Solve for k. The value of k = -4. The equation is y = -4x

Writing and Using a Direct Variation Equation PROBLEM: The variables x and y vary directly, and x = -3 when y = 12. Write an equation that relates the variables. Then find y when x = -2. SOLUTION y = -4x Write direct variation equation. y = -4(-2) Substitute (-2) for x. y = 8 Solve for y. The value of y = 8 when x = -2.

ASSIGNMENT pg. 95-98. #13, #17, #19, #23, #25, #27, #29, #33, #35, & #39.