Linear Graphs | Mathematics for GCSE/IGCSE - Year 11 PDF Download

Finding Equations of Straight Lines

Why do we want to know about straight lines and their equations?

• Straight Line Graphs (Linear Graphs) have numerous applications in mathematics, such as in navigation.
• We might need to determine the equation of a straight line to program it into a computer. This computer can then plot the line on a screen, alongside other lines, to create shapes and graphics.

How do we find the equation of a straight line?

• The equation of a straight line is defined by the formula: (y = mx + c)
  • Here, ( m ) represents the gradient, and ( c ) is the y-axis intercept (or simply, the y-intercept)
• To determine the equation of a straight line, you need two essential pieces of information:
  • Obtain the gradient, ( m), directly from the question or calculate it using or gradient formula or two given points
  • Choose any point on the line and substitute its coordinates into y = mx + c (as you already know both ( m ) and the point)
  • Solve to find the value of ( c ) if you are given two points that were used to determine the gradient
• If ( m ) is a fraction, you may be required to present the equation in a suitable form ax + by + c = 0
• When in doubt, sketch the line to visually represent the problem

What if the line is not in the form y = mx + c?

• A line could be given in the form ax + by + c = 0
  • It is harder to identify the gradient and intercept in this form
• We can rearrange the equation into y = mx + c, so it is easier to identify the gradient and intercept
  • ax + by + c = 0
  • Subtract ax from both sides
    • by + c = - ax
  • Subtract c from both sides
    • by = - ax - c
  • Divide both sides by b
    • 
    • 
  • In this case, the gradient is and the y-intercept is

Drawing Linear Graphs

Drawing Linear Graphs: Understanding the Basics 

• Before attempting to sketch a straight line, it's crucial to grasp how to derive the equation of a straight line. This understanding is fundamental to drawing accurate graphs.
• The method of drawing a straight line varies based on the form in which the equation is presented. The two primary forms are y = mx + c and ax + by = c.

Different Approaches to Drawing Linear Graphs 

• When the equation is in the form y = mx + c, where 'm' is the slope and 'c' is the y-intercept, follow these steps:
  • Locate 'c' on the y-axis.
  • Move 1 unit to the right and 'm' units upwards (repeat this process to plot more points on the line).
• For the equation ax + by = c, you can identify the intercepts by:
  • Setting x = 0 to find the y-axis intercept.
  • Setting y = 0 to find the x-axis intercept. 

(Alternatively, you may rearrange the equation to y = mx + c and apply the previous method.)