Equations of a Straight Line (DP IB Maths: AI HL)

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Dan

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Dan

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Equations of a Straight Line

How do I find the gradient of a straight line?

  • Find two points that the line passes through with coordinates (x1, y1) and (x2, y2)
  • The gradient between these two points is calculated by

begin mathsize 20px style m equals fraction numerator y subscript 2 minus y subscript 1 over denominator x subscript 2 minus x subscript 1 end fraction end style 

    • This is given in the formula booklet
  • The gradient of a straight line measures its slope
    • A line with gradient 1 will go up 1 unit for every unit it goes to the right
    • A line with gradient -2 will go down two units for every unit it goes to the right

What are the equations of a straight line?

  • space y equals m x plus c
    • This is the gradient-intercept form
    • It clearly shows the gradient m and the y-intercept (0, c)
  • space y minus y subscript 1 equals m open parentheses x minus x subscript 1 close parentheses
    • This is the point-gradient form
    • It clearly shows the gradient m and a point on the line (x1, y1)
  • space a x plus b y plus d equals 0
    • This is the general form
    • You can quickly get the x-interceptspace stretchy left parenthesis negative d over a comma space 0 stretchy right parenthesis and y-interceptspace stretchy left parenthesis 0 comma space minus d over b stretchy right parenthesis

How do I find an equation of a straight line?

  • You will need the gradient
    • If you are given two points then first find the gradient
  • It is easiest to start with the point-gradient form
    • then rearrange into whatever form is required
      • multiplying both sides by any denominators will get rid of fractions
  • You can check your answer by using your GDC
    • Graph your answer and check it goes through the point(s)
    • If you have two points then you can enter these in the statistics mode and find the regression line space y equals a x plus b

Examiner Tip

  • A sketch of the graph of the straight line(s) can be helpful, even if not demanded by the question
    • Use your GDC to plot them
  • Ensure you state equations of straight lines in the format required
    • Usually  y equals m x plus c  or  a x plus b y plus d equals 0
    • Check whether coefficients need to be integers (they usually are for a x plus b y plus d equals 0)

Worked example

The line space l passes through the points left parenthesis negative 2 comma space 5 right parenthesis and left parenthesis 6 comma space minus 7 right parenthesis.

Find the equation of space l , giving your answer in the form space a x plus b y plus d equals 0 where space a comma space b  and space d are integers to be found.

2-1-1-ib-ai-sl-equation-of-a-line-we-solution

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Parallel Lines

How are the equations of parallel lines connected?

  • Parallel lines are always equidistant meaning they never intersect
  • Parallel lines have the same gradient
    • If the gradient of line l1 is m1 and gradient of line l2 is mthen...
      • m subscript 1 equals m subscript 2 rightwards double arrow l subscript 1 blank & blank l subscript 2 blank are blank parallel
      • l subscript 1 blank & blank l subscript 2 blank are blank parallel rightwards double arrow m subscript 1 equals m subscript 2
  • To determine if two lines are parallel:
    • Rearrange into the gradient-intercept form space y equals m x plus c
    • Compare the coefficients of space x
    • If they are equal then the lines are parallel

Parallel & Perpendicular Gradients Notes Diagram 1

Worked example

The line space l  passes through the point space left parenthesis 4 comma negative 1 right parenthesis  and is parallel to the line with equation space 2 x minus 5 y equals 3 .

Find the equation of space l , giving your answer in the form space y equals m x plus c.

2-1-1-ib-ai-sl-parallel-lines-we-solution 

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Perpendicular Lines

How are the equations of perpendicular lines connected?

  • Perpendicular lines intersect at right angles
  • The gradients of two perpendicular lines are negative reciprocals
    • If the gradient of line l1 is m1 and gradient of line l2 is mthen...
      • space m subscript 1 cross times m subscript 2 equals negative 1 space rightwards double arrow space l subscript 1 space & space l subscript 2 space are space perpendicular
      •   l subscript 1 space & space l subscript 2 space are space perpendicular space rightwards double arrow space m subscript 1 cross times m subscript 2 equals negative 1
  • To determine if two lines are perpendicular:
    • Rearrange into the gradient-intercept form space y equals m x plus c
    • Compare the coefficients of space x
    • If their product is -1 then they are perpendicular
  • Be careful with horizontal and vertical lines
    • space x equals p and space y equals q are perpendicular where p and q are constants

Parallel & Perpendicular Gradients Notes Diagram 3

Worked example

The line space l subscript 1  is given by the equation space 3 x minus 5 y equals 7.

The line space l subscript 2  is given by the equation space y equals 1 fourth minus 5 over 3 x .

Determine whether space l subscript 1 and space l subscript 2 are perpendicular. Give a reason for your answer.

2-1-1-ib-ai-sl-perpendicular-lines-we-solution

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Dan

Author: Dan

Expertise: Maths

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.