2.1.1 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

 

• • 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?

• 
  • This is the gradient-intercept form
  • It clearly shows the gradient m and the y-intercept (0, c)
• 
• • This is the point-gradient form
  • It clearly shows the gradient m and a point on the line (x₁, y₁)
• 
  • This is the general form
  • You can quickly get the x-intercept and y-intercept

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 

Worked Example

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 m₂ then...
    • 
    • 
• To determine if two lines are parallel:
  • Rearrange into the gradient-intercept form 
  • Compare the coefficients of 
  • If they are equal then the lines are parallel

Worked Example

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 m₂ then...
    • 
    •  
• To determine if two lines are perpendicular:
  • Rearrange into the gradient-intercept form 
  • Compare the coefficients of 
  • If their product is -1 then they are perpendicular
• Be careful with horizontal and vertical lines
  • and  are perpendicular where p and q are constants

Worked Example