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