3 1 lines

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3 1 lines

  1. 1. Lines
  2. 2. <ul><li>Linear equation – a polynomial equation of the first degree </li></ul><ul><li>The term “linear” stems from the fact that the graph of such an equation is a straight line. </li></ul><ul><li>Forms: </li></ul>
  3. 3. <ul><li>To graph a line, we generate a pair of points using its equation and then connect the two points. </li></ul><ul><li>Example. If the linear equation is , a pair of points is generated when we let x = 0 and then x = 1 . </li></ul>
  4. 4. <ul><li>Connecting the two points (0,1) and give us the graph of the line. </li></ul>
  5. 5. <ul><li>Special cases: </li></ul><ul><li>y = b </li></ul><ul><li>x = a </li></ul>
  6. 6. <ul><li>The slope of a line describes its incline. </li></ul><ul><li>The higher the value of the slope, the steeper the incline is. </li></ul><ul><li>The slope is also defined as a rate of change (the ratio of the change in y coordinate to the change in x coordinate between any two points on the line). </li></ul>
  7. 7. <ul><li>If the line is not vertical and ( x 1 , y 1 ) and ( x 2 , y 2 ) are distinct points on the line, then the slope of the line is </li></ul>
  8. 8. <ul><li>Example. Find the slope of the line that passes through the points ( - 1,0) and (3,8) . </li></ul><ul><li>The slope m is given by </li></ul>
  9. 9. <ul><li>The slope of a horizontal line is zero while that of a vertical line is not defined. </li></ul><ul><li>Two non-vertical lines are parallel if and only if m 1 = m 2 . </li></ul><ul><li>Two lines are perpendicular if and only if m 1 m 2 = - 1 . </li></ul>
  10. 10. <ul><li>Example. What is the slope of the line parallel to the line whose equation is ? </li></ul><ul><li>Rewrite the given equation into the form y = mx + b . The slope is the coefficient m of x . </li></ul><ul><li>Hence, the slope of the given line is 2. </li></ul><ul><li>Since two parallel lines have equal slopes, the other line must also have a slope of 2. </li></ul>
  11. 11. <ul><li>Linear equations can be rewritten into several different forms. These forms are collectively referred to as “equations of the straight line”. </li></ul><ul><li>Slope-intercept form: y = mx + b , where m is the slope and b is the y -intercept </li></ul><ul><li>Illustration. The equation of the line with slope 2 and y -intercept –5 is </li></ul>
  12. 12. <ul><li>Two-point form: , with . the line passing through the points ( x 1 , y 1 ) and ( x 2 , y 2 ) </li></ul><ul><li>Illustration. The equation of the line which passes through (2,1) and (-1,5) is </li></ul>
  13. 13. <ul><li>Point-slope form: y – y 1 = m ( x – x 1 ) , with the line having a slope m and passing through the point ( x 1 , y 1 ) </li></ul><ul><li>Illustration. The equation of the line whose slope is and passes through (3,5) is </li></ul>
  14. 14. <ul><li>Intercept form: , with x -intercept a and . y -intercept b </li></ul><ul><li>Illustration. The equation of the line with y -intercept 5 and x -intercept - 1 is </li></ul>
  15. 15. <ul><li>1.a Graph the line with slope and passing through the point (1,4) . </li></ul><ul><li>From (1,4), move 2 units up and then 3 units to the right. Connect the points. </li></ul>
  16. 16. <ul><li>1.b Graph the line with slope and passing through the point (2,5) . </li></ul><ul><li>From (2,5), move 1 unit up and then 4 units to the left (or 1 unit down and then 4 units to the right) </li></ul>
  17. 17. <ul><li>2.a Find the slope and y - intercept of the line 2 x – y = 4 . </li></ul>
  18. 18. <ul><li>2.d Find the slope and y - intercept of the line 3( y + 1) = 2( x – 5) . </li></ul>
  19. 19. <ul><li>3.a Find an equation of the line passing through (4,3) and (2,5) . Express your answer in slope-intercept form. </li></ul>
  20. 20. <ul><li>3.d Find an equation of the line passing through (3,2) and has slope 3 . Express your answer in slope-intercept form. </li></ul>
  21. 21. <ul><li>3.g Find an equation of the line with slope 4 and y -intercept 2 . Express your answer in slope-intercept form. </li></ul>
  22. 22. <ul><li>4.a Graph . </li></ul><ul><li>Draw a line through (0,200) and (1,225). </li></ul>
  23. 23. <ul><li>5.a Find an equation of the line passing through ( - 2,4) and is perpendicular to the line 4 x + 3 y = 2 . Express your answer in slope-intercept form. </li></ul>
  24. 24. <ul><li>5.c Find an equation of the line passing through (1,4) and is parallel to the line - 4 x + 6 y = 2 . Express your answer in slope-intercept form. </li></ul>
  25. 25. <ul><li>5.g Find an equation of the line passing through (4, - 3) and has a slope of 0 . Express your answer in slope-intercept form. </li></ul><ul><li>If the slope is 0, the line is horizontal. </li></ul><ul><li>So our line is a horizontal line passing through (4, - 3) . </li></ul><ul><li>The equation is y = –3 . </li></ul>
  26. 26. <ul><li>6.a Are the following pairs of lines parallel, perpendicular, or neither? </li></ul>

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