Chapter 22

Using the Algebra of Lines

The equation of a line can take on one of many forms. The more popular are the slope-intercept form, y = mx + b, and the standard form, Ax + By = C. Each has its advantages and uses in the different applications for lines. Formulas involving lines are often based on one of these forms.

The Problems You'll Work On

Lining up equations and applications in this chapter requires the following techniques:

  • Writing the equation of a line given slope and a point
  • Writing the equation of a line given two points
  • Determining the slopes of lines parallel or perpendicular to a particular line
  • Writing the equations of lines parallel or perpendicular to a particular line through a particular point
  • Finding the distance between points
  • Finding the midpoint of a segment between two points

What to Watch Out For

Although working with line formulas is the main skill in this chapter, don't forget the following:

  • Calculating the slope correctly when given two points
  • Remembering to use the negative reciprocal when determining slopes of perpendicular lines
  • Calculating squares and square roots correctly and using order of operations correctly in the distance formula
  • Adding signed numbers correctly when determining midpoints

Writing Equations of Lines Using the Slope and a Point

926−931 Write the equation of the line given the slope and a point on the line.

926. m = 2, (−3, 4)

927. image

928. m = −3, (4, 0)

929. image

930. m = 0, (4, 3)

931. m is undefined, (−3, 5)

Writing the Equation of a Line Using Two Points

932−937 Write the equation of the line that passes through the two points.

932. (4, −1) and (6, −5)

933. (3, 3) and (−5, −7)

934. (1, 6) and (3, 6)

935. (4, −3) and (0, 2)

936. (−4, 5) and (−4, −5)

937. (0, 0) and (3, −8)

Dealing with Slopes of Parallel and Perpendicular Lines

938−943 Determine the slopes of the lines parallel and perpendicular to the given line.

938. y = 4x − 3

939. image

940. 2x − 3y = 7

941. x − 4y = 8

942. x = 5

943. y = −6

Finding Equations of Lines Parallel or Perpendicular to One Given

944−947 Find the equations of lines parallel and perpendicular to the line through the given point.

944. y = −2x + 1 through (0, 3)

945. image

946. 4x − y = 3 through (0, 0)

947. 6x + 3y = 7 through (−1, 1)

Computing the Distance Between Points

948−957 Find the distance between the two points.

948. (3, 4) and (−2, −8)

949. (1, −3) and (−5, 5)

950. (−4, −3) and (0, 0)

951. (5, −2) and (−2, 22)

952. (3, 3) and (−2, −2)

953. (−4, 1) and (6, 9)

954. (−3, 7) and (0, −2)

955. (0, 4) and (4, 0)

956. (−3, 6) and (−3, 8)

957. (−5, 4) and (5, −4)

Determining the Midpoint

958−965 Find the midpoint of the two points.

958. (4, 7) and (2, −5)

959. (−3, 6) and (−5, −4)

960. (1, 6) and (−3, −2)

961. (−6, −6) and (−8, 0)

962. (4, −3) and (5, 9)

963. (−3, 8) and (−3, 10)

964. (4, 0) and (0, 0)

965. image

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