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

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

$m=\frac{y_2-y_1}{x_2-x_1}$ 

• 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?

• $y=mx+c$

  • This is sometimes called gradient-intercept form

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

• $y-y_1=m(x-x_1)$

  • This is sometimes called the point-gradient form

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

• $ax+by=c$

  • This is sometimes called the general form

  • You can quickly get the x-intercept$(\frac{c}{a},0)$,  y-intercept$(0,\frac{c}{b})$  and gradient $m=-\frac{a}{b}$

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

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

    • $y=-\frac{a}{b}x+\frac{c}{b}$

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 $y-y_1=m\left(x-x_1\right)$

  • 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 $l_1$ is $m_1$ and the gradient of line $l_2$ is $m_2$ then

    • If $m_1=m_2$ then $l_1$ and $l_2$ are parallel

    • If $l_1$ and $l_2$ are parallel, then $m_1=m_2$

• To determine if two lines are parallel:

  • Rearrange into the form $y=mx+c$

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

  • 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 $-1$

    • e.g. $\frac{1}{2}$ and $-2$

  • If the gradient of line $l_1$ is $m_1$ and the gradient of line $l_2$ is $m_2$ then...

    • If $m_1\times m_2=-1$ then $l_1$ and $l_2$ are perpendicular

    • If $l_1$ and $l_2$ are perpendicular, then $m_1\times m_2=-1$

• To determine if two lines are perpendicular:

  • Rearrange into the form $y=mx+c$

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

  • If their product is $\mathbf{-}\mathbf{1}$ then the lines are perpendicular

• Be careful with horizontal and vertical lines

  • $x=p$ and $y=q$ are perpendicular where p and q are constants