Equation of a Straight Line (AQA GCSE Further Maths)

What is the equation of a straight line?

• y = mx + c is the equation for any straight line
• m is gradient given by “difference in y” ÷ “difference in x” or
• c is the y-axis intercept
• An alternative form is ax + by + c = 0
  • where a, b and c are integers

How do I find the equation of a straight line?

•  Two features of a straight line are needed
  • gradient, m
  • a point the line passes through, (x₁, y₁)

• The equation can then be found using y – y₁ = m(x - x₁)
• This can be arranged into either y = mx + c or ax + by + c = 0

How do I find the gradient of a straight line?

• There are lots of ways to find the gradient of a line

• Using two points on a line to find the change in y divided by change in x

• Using the fact that lines are parallel or perpendicular to another line
  • see Gradients - Parallel and Perpendicular Gradients

• Using Tangents and Normals - Differentiation
  • see Tangents & Normals

• Other ways
  • Collinear lines are the same straight line so gradients are equal
  • Angle facts and circle theorems
    e.g. a radius and tangent are perpendicular

How do I find where two lines intersect?

• The coordinates of the point of intersection of two lines, , is the solution for  and when the pair of equations are solved simultaneously
  • A pair of parallel lines will not have any intersections, and therefore no solutions to the simultaneous equation, so it is useful to check if the lines are parallel before trying to solve them
• We can find the solutions using either substitution or elimination
• The most straight-forward method for finding where two straight lines intersect is to
  • Rearrange each line's equation to
  • Substitute one expression for  into the other equation, in place of 
  • Solve to find
  • Substitute the value of into either line's equation to find
• The method is exactly the same when finding where a line and a curve, or two curves, intersect
  • When finding where a quadratic curve and a line intersect, you will end up with a quadratic equation to solve, and there could be either 2, 1, or 0 solutions
    • This corresponds to the number of intersections of the line and the curve
• The same idea can be used to find the intersection of a line or curve with the  or  axis
  • The equation of the -axis is 
  • The equation of the -axis is 
  • Either of these can be substituted into the equation of the line or curve and solved

What are parallel lines?

• Parallel lines are lines that have the same gradient, but are not the same line
• Parallel lines do not intersect with each other
• You can easily spot that two lines are parallel when they are written in the form , as they will have the same value of  (gradient)
•  and  are parallel
•  and  are not parallel
•  and  are not parallel; they are the exact same line

How do I find the equation of a line parallel to another line?

• As parallel lines have the same gradient, a line of the form  will be parallel to a line in the form , where  is the same for both lines
• If  then they would be the same line and therefore not parallel
• If you are asked to find the equation of a line parallel to , you will also be given some information about a point that the parallel line, passes through; 
• You can then substitute this point into and solve to find 

What are perpendicular lines?

• You should already know that parallel lines have equal gradients
• Perpendicular lines meet each other at right angles
  • i.e. they meet at 90°

What’s the deal with perpendicular gradients (and lines)?

• Before you start trying to work with perpendicular gradients and lines, make sure you understand how to find the equation of a straight line – that will help you do the sorts of questions you will meet
• Gradients m₁ and m₂ are perpendicular if m₁ × m₂ = −1
  • For example
    • 1 and -1
    • and -3
    • and 
• We can use m₂ = −1 ÷ m₁ to find a perpendicular gradient. This is called the negative reciprocal.

• If in doubt, SKETCH IT!