Equations of a Straight Line (Edexcel IGCSE Further Pure Maths)

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 m  between these two points is calculated by

• This is sometimes known as rise over run

• 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 sometimes called gradient-intercept form

  • It clearly shows the gradient m and the y-intercept (0, c)

• 

  • This is sometimes called the point-gradient form

  • It clearly shows the gradient m and a point on the line (x₁, y₁)

• 

  • This is sometimes called the general form

  • You can quickly get the x-intercept,  y-intercept  and gradient 

    • is not the y-intercept in this form of the line equation!

  • You can also rearrange the equation into gradient-intercept form

    • 

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

• Always check your answer

  • Substitute the coordinates of points on the line into the equation 

  • Make sure the equation is satisfied with those coordinates

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 is and the gradient of line is  then

    • If then  and  are parallel

    • If  and  are parallel, then 

• To determine if two lines are parallel:

  • Rearrange into the form

  • Compare the gradients (i.e. the coefficients of )

  • If they are equal then the lines are parallel

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

  • This means the product of their gradients is equal to 

    • e.g.  and 

  • If the gradient of line is and the gradient of line is  then...

    • If  then  and  are perpendicular

    • If  and  are perpendicular, then 

• To determine if two lines are perpendicular:

  • Rearrange into the form

  • Compare the gradients (i.e. the coefficients of)

  • If their product is then the lines are perpendicular

• Be careful with horizontal and vertical lines

  • and  are perpendicular where p and q are constants