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Equations of Lines

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Equations of Lines

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  1. Point-slope form: y – y1 = m(x – x1) • Slope-intercept form: y = mx + b • Horizontal line: y = b • Vertical line: x = a • General form: Ax + By + C = 0 • Standard form: Ax + By = C Equations of Lines

  2. The slope-intercept equation of a non-vertical line with slope m and y-intercept b is y = mx + b. Slope-Intercept Form of the Equation of a Line Point-Slope Form of the Equation of a Line The point-slope equation of a non-vertical line of slope m that passes through the point (x1, y1) is y – y1 = m(x – x1).

  3. Equation of a Horizontal Line A horizontal line is given by an equation of the form y = b where b is the y-intercept. Note: m = 0. Equations of Horizontal and Vertical Lines Equation of a Vertical Line A vertical line is given by an equation of the form x = a where a is the x-intercept. Note: m is undefined.

  4. y – y1 = m(x – x1) This is the point-slope form of the equation. y – 3 = 4[x – (-1)] Substitute the given values. Simply. We now have the point-slope form of the equation for the given line. y – 3 = 4(x + 1) y = 4x + 7 Add 3 to both sides. Write the point-slope form of the equation of the line passing through (-1,3) with a slope of 4. Then solve the equation for y. Example: Writing the Point-Slope Equation of a Line SolutionWe use the point-slope equation of a line with m = 4, x1= -1, and y1 = 3. We can solve the equation for y by applying the distributive property. y – 3 = 4x + 4

  5. If two non-vertical lines are parallel, then they have the same slope. • If two distinct non-vertical lines have the same slope, then they are parallel. • Two distinct vertical lines, both with undefined slopes, are parallel. Slope and Parallel Lines

  6. 90° Two lines that intersect at a right angle (90°) are said to be perpendicular. There is a relationship between the slopes of perpendicular lines. Slope and Perpendicular Lines • Slope and Perpendicular Lines • If two non-vertical lines are perpendicular, then the product of their slopes is –1. • If the product of the slopes of two lines is –1, then the lines are perpendicular. • A horizontal line having zero slope is perpendicular to a vertical line having undefined slope.

  7. Graphing y = mx + b by Using the Slope and y-Intercept • Plot the y-intercept on the y-axis. This is the point (0, b). • Obtain a second point using the slope, m. Write m as a fraction, and use rise over run starting at the y-intercept to plot this point. • Use a straightedge to draw a line through the two points. Draw arrowheads at the ends of the line to show that the line continues indefinitely in both directions. Steps for Graphing y = mx + b

  8. y = x + 2 The slope is 2/3. The y-intercept is 2. more more Graph the line whose equation is y = x + 2. Example: Graphing by Using Slope and y-Intercept SolutionThe equation of the line is in the form y = mx + b. We can find the slope, m, by identifying the coefficient of x. We can find the y-intercept, b, by identifying the constant term.

  9. Solution We need two points in order to graph the line. We can use the y-intercept, 2, to obtain the first point (0, 2). Plot this point on the y-axis. 5 4 3 2 1 -5 -4 -3 -2 -1 1 2 3 4 5 -1 -2 -3 -4 -5 Graph the line whose equation is y = x + 2. Example: Graphing by Using Slope and y-Intercept We plot the second point on the line by starting at (0, 2), the first point. Then move 2 units up (the rise) and 3 units to the right (the run). This gives us a second point at (3, 4).

  10. 2x – 3y + 6 = 0 This is the given equation. 2x + 6 = 3y To isolate the y-term, add 3 y on both sides. 3y = 2x + 6 Reverse the two sides. (This step is optional.) Divide both sides by 3. Find the slope and the y-intercept of the line whose equation is 2x – 3y + 6 = 0. Example: Finding the Slope and the y-Intercept SolutionThe equation is given in general form, Ax + By + C = 0. One method is to rewrite it in the form y = mx + b. We need to solve for y. The coefficient of x, 2/3, is the slope and the constant term, 2, is the y-intercept.

  11. 2-4 WRITING LINEAR EQUATIONS

  12. Equations of the Line • Write the equation of a line given the slope and the y-intercept • Write the equation of a line given the slope and a point • Write the equation of a line given two points

  13. Equations of the Line • Write the equation of a line given the slope and the y-intercept: m and (0, b) • Write the equation of a line given the slope and a point • Write the equation of a line given two points

  14. Ex: Find an equation of the line with slope = 6 and y-int = (0, -3/2) Recall: slope-intercept form of a linear equation y = mx + b, where m and b are constants Given the y-int = (0, -3/2)  b = - 3/2 Given the slope = 6 m = 6 Putting everything together we get the equation of the line in slope-int form: y m 6 x + b - 3/2 = y = 6x – 3/2

  15. Ex: Find an equation of the line with slope = 1.23 and y-int = (0, 0.63) Recall: slope-intercept form of a linear equation y = mx + b, where m and b are constants Given the y-int = (0, 0.63)  b = 0.63 Given the slope = 1.23 m = 1.23 Putting everything together we get the equation of the line in slope-int form: y m + b = 1.23 x 0.63 y = 1.23x + 0.63

  16. Equations of the Line • Write the equation of a line given the slope and the y-intercept • Write the equation of a line given the slope and a point: m and (x1, y1) • Write the equation of a line given two points

  17. Ex: Find an equation of the line with slope = -3 that contains the point (4, 2) Start with the slope-intercept form of a linear equation y = mx + b Slope = - 3 y = - 3x + b What is b, though? What is b, though? Use the given point (4, 2) x = 4 and y = 2 y = - 3x + b  2 = - 3(4) + b put it together 2 = -12 + b we have m and b 14 = b y = - 3 x + 14

  18. Ex: Find an equation of the line with slope = -0.25 that contains the point (2, -6) Start with the slope-intercept form of a linear equation y = mx + b Slope = -0.25  y = -0.25 x + b What is b, though? Use the given point (2, -6) x = 2 and y = -6 y = -0.25 x + b  -6 = -0.25(2) + b put it together -6 = -0.5 + b we have m and b -5.5 = b y = -0.25x – 5.5

  19. Equations of the Line • Write the equation of a line given the slope and the y-intercept • Write the equation of a line given the slope and a point • Write the equation of a line given two points: (x1, y1) and (x2, y2)

  20. Ex: Find an equation of the line containing the points (-2, 1) and (3, 5) Point 1 Point 2 First, find the slope of the line containing the points: Slope = m = rise = y1 - y2= 1 – (5) run x1 - x2 -2 – 3 Now we have m = 4/5 and two points. Pick one point and proceed like in the last section.

  21. We have m = 4/5, the point (-2, 1), and y = mx + b Slope = 4/5  y = 4/5x + b What is b, though? Use the given point (-2, 1) x = -2 and y = 1 y = 4/5x + b  1 = 4/5(-2) + b 1 = (-8/5) + b 13/5 = b put it together  we have m and b y = 4/5x + 13/5

  22. Ex: Find an equation of the line containing the points (-4, 5) and (-2, -3) Point 1 Point 2 First, find the slope of the line containing the points: Slope = m = rise = y1 - y2= 5 – (-3) run x1 - x2 -4 – (-2) = -4 Now we have m = -4 and two points. Pick one point and proceed like in the last section.

  23. We have m = -4, the point (-4, 5), and y = mx + b Slope = -4  y = -4x + b What is b, though? Use the given point (-4, 5) x = -4 and y = 5 y = -4x + b  5 = -4(-4) + b 5 = 16 + b -11 = b put it together  we have m and b y = -4x – 11

  24. Ex: Find an equation of the line containing the points (0, 0) and (1, -5) Point 1 Point 2 First, find the slope of the line containing the points: Slope = m = rise = y1 - y2= 0 – (-5) run x1 - x2 0 – (1) = -5 Now we have m = -5 and two points. Pick one point and proceed like in the last section.

  25. We have m = -5, the point (0, 0), and y = mx + b Slope = -5  y = -5x + b What is b, though? Use the given point (0, 0) x = 0 and y = 0 y = -5x + b  0 = -5(0) + b 0 = 0 + b 0 = b put it together  we have m and b y = -5x + 0 y = -5x

  26. Equations of the Line • Write the equation of a line given the slope and the y-intercept • Write the equation of a line given the slope and a point • Write the equation of a line given two points

  27. Parallel & Perpendicular Lines • When we graph a pair of linear equations, there are three possibilities: • the graphs intersect at exactly one point • the graphs do not intersect • the graphs intersect at infinitely many points • We will consider a special case of situation 1 and also situation 2.

  28. Perpendicular Lines (Situation 1) • Perpendicular lines intersect at a right angle • Notation: • L1: y = m1x + b1 • L2: y = m2x + b2 • L1^ L2

  29. Nonvertical perpendicular lines have slopes that are the negative reciprocals of each other: m1m2 = -1 ~ or ~ m1 = - 1/m2~ or ~ m2 = - 1/m1 If l1 is vertical (l1: x = a) and is perpendicular to l2, then l2 is horizontal (l2: y = b) ~ and ~ vice versa

  30. Ex: Determine whether or not the graphs of the equations of the lines are perpendicular:l1: x + y = 8 and l2: x – y = - 1 First, determine the slopes of each line by rewriting the equations in slope-intercept form: l1: y = - x + 8 and l2: y = x + 1 1 1 m1 = and m2 = -1 1 Since m1m2 = (-1)(1) = -1, the lines are perpendicular.

  31. Ex: Determine whether or not the graphs of the equations of the lines are perpendicular:l1: -2x + 3y = -21 and l2: 2y – 3x = 16 First, determine the slopes of each line by rewriting the equations in slope-intercept form: l1: y = (2/3)x - 7 and l2: y = (3/2)x + 8 m1 = and m2 = 2/3 3/2 Since m1m2 = (2/3)(3/2) = 1 = -1 Therefore, the lines are not perpendicular!

  32. Parallel Lines (Situation 2) • Parallel lines do not intersect • Notation: • L1: y = m1x + b1 • L2: y = m2x + b2 • L1|| L2

  33. Nonvertical parallel lines have the same slopes but different y-intercepts: m1 = m2~ and ~ b1 = b2 Horizontal Parallel Lines have equations y = p and y = q where p and q differ. Vertical Parallel Lines have equations x = p and x = q where p and q differ.

  34. Ex: Determine whether or not the graphs of the equations of the lines are parallel:l1: 3x - y = -5 and l2: y – 3x = - 2 First, determine the slopes and intercepts of each line by rewriting the equations in slope-intercept form: l1: y = 3x + 5 and l2: y = 3x - 2 m1 = and m2 = 3 3 b1 = and b2 = 5 -2 Since m1 = m2 and b1 = b2 the lines are parallel.

  35. Ex: Determine whether or not the graphs of the equations of the lines are parallel:l1: 4x + y = 3 and l2: x + 4y = - 4 First, determine the slopes and intercepts of each line by rewriting the equations in slope-intercept form: l1: y = -4x + 3 and l2: y = (-¼)x - 1 m1 = and m2 = -4 - ¼ Since m1 = m2 the lines are not parallel.

  36. HW: 2-4 WORKSHEET Front and Back (Due Next Time)