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Straight Line Graphs (y = mx + c) (CIE IGCSE Maths: Core)

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Daniel I

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Finding Equations of Straight Lines

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

  • Straight Line Graphs (Linear Graphs) have lots of uses in mathematics – one use is in navigation
  • We may want to know the equation of a straight line so we can program it into a computer that will plot the line on a screen, along with several others, to make shapes and graphics

How do we find the equation of a straight line?

  • The general equation of a straight line is bold italic y bold equals bold italic m bold italic x bold plus bold italic c where;
    • bold italic m is the gradient,
    • bold italic c is the y-axis intercept (or simply, the y-intercept)
  • To find the equation of a straight line you need TWO things:
    1. the gradient, m, which you can put straight into y equals m x plus c
      • get this from the question directly, or from a diagram
    2. any point on the line- substitute this point into y equals m x plus c (as you already know m) and solve to find c
      • if given two points which you used to find the gradient, just choose either one of them for the point to find c
  • You may be asked to give the equation in the form a x plus b y plus c equals 0

    (especially if m is a fraction)

    If in doubt, SKETCH IT!

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

  • A line could be given in the form a x plus b y plus c equals 0
    • It is harder to identify the gradient and intercept in this form
  • We can rearrange the equation into y equals m x plus c, so it is easier to identify the gradient and intercept
    • a x plus b y plus c equals 0
    • This is easiest done when the values of a comma space b and c are known
      • e.g.  The equation 2 x plus 5 y minus 6 equals 0 can be rearranged to
        5 y equals negative 2 x plus 6
y equals negative 2 over 5 x plus 6 over 5
      • So the gradient is negative 2 over 5 and the line intercepts the y-axis at the point open parentheses 0 comma 6 over 5 close parentheses

Worked example

(a)

Find the equation of the straight line with gradient 3 that passes through (5, 4).


We know that the gradient is 3 so the line takes the form 

y equals 3 x plus c

To find the value of c, substitute (5, 4) into the equation


4 equals 3 open parentheses 5 close parentheses plus c
4 equals 15 plus c
c equals negative 11


Replace c with −11 to complete the equation of the line

y = 3x − 11

(b)
Find the equation of the straight line shown in the diagram below.
screenshot-2023-02-13-at-10-14-19


First find m, the gradient
Identify two points the line passes through and work out the rise and run

Line passes through (2, 4) and (10, 0)

cie-igce-core-rn-finding-equations-of-straight-lines-we-solution-1

∴ rise = 4    (4 - 0)
     run = 8    (10 - 2)

therefore rise over run equals 4 over 8 equals 1 half

!! Remember we also need to consider whether the line is "uphill" or "downhill" to find the gradient !!
The slope is downward in the left to right sense, so it is a negative gradient

∴ gradient, m equals negative 1 half

We now know that the line takes the form 
table row y equals cell negative 1 half x plus c end cell end table

To find the value of c, substitute one of the points identified earlier into this equation
Here we've used (10, 0) as 0's often make the maths easier

0 equals negative 1 half open parentheses 10 close parentheses plus c
0 equals negative 5 plus c
c equals 5


Replace c with 5 to complete the equation of the line

bold italic y bold equals bold minus bold 1 over bold 2 bold italic x bold plus bold 5

We can check against our sketch that this equation looks correct- it has a negative gradient and it crosses the y-axis at 5.

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Drawing Linear Graphs

How do we draw the graph of a straight line from an equation?

  • Before you start trying to draw a straight line, make sure you understand how to find the equation of a straight line – that will help you understand this
  • How we draw a straight line depends on what form the equation is given in
  • There are two main forms you might see:

    y = mx + c and ax + by + c = 0

  • Different ways of drawing the graph of a straight line:
  1. From the form y = mx + c
    (you might be able to rearrange to this form easily)


    plot c on the y-axis

    go 1 across, m up (and repeat until you can draw the line)

  2. From ax + by + c = 0

    put x = 0 to find y-axis intercept

    put y = 0 to find x-axis intercept

    (You may prefer to rearrange to y = mx + c and use above method)
     
  3. Work out and plot two points on the line

    Pick a value of x, substitute this into the equation of the line (whichever form it is in) to find the corresponding y coordinate

Repeat this for a different value of x

Plot the two points and join them up

It is often a good idea to pick and plot a third point, as a check
All three points will be on the line if they're correct - if not, at least one of the points is incorrect!

How do we draw the graph of a straight line from a table of values?

  • Some questions will not give you a diagram, nor tell you a graph is a straight line (linear) graph
  • Instead they will give you a table of x and y values to complete using the equation of the straight line
    • The line is then drawn using the values in the table as coordinates
  • If you recognise that the equation (or pattern in the values) will give a straight line graph then you will know if any points are incorrect when you plot them
  • For example, complete the table of value below for the equation y equals negative 2 x plus 6

    x -3 -2 -1 0 1 2 3
    y   2       4  


    Substitute each x value in turn to find the corresponding y value

    y equals negative 2 cross times open parentheses negative 3 close parentheses plus 6 equals 6 plus 6 equals 12

    and so on for the other missing y values

    Then you would plot each coordinate and join them up to draw the straight line graph.
    If any coordinates do not lie on the line you should go back and check your calculations for that one.

Examiner Tip

  • If plotting a straight line from a table of values then there is no need to plot them all
    • You'll need to plot at least 2 points
    • We recommend you plot a third too as a way of checking the first two
  • To complete a table of values, you can use the TABLE feature/mode of your calculator, if it has one
    • Most modern calculators have this feature but it may be hidden under on screen menus rather than having a dedicated key

Worked example

On the axes below, draw the graphs of y equals 3 x minus 1 and 3 x plus 5 y equals 15.


For y equals 3 x minus 1, first plot c, which is (0, −1)

Then, as m = 3, rise over run equals 3 over 1. So plot a point 3 up and 1 right. Repeat at least once more and then join the points with a straight line. Extend the line to the edges of the grid.

The steps for 3 x plus 5 y equals 15 are the same, but first we need to rearrange into the form y = mx + c

table row cell 5 y end cell equals cell negative 3 x plus 15 end cell row y equals cell negative 3 over 5 x plus 3 end cell end table

Now we can plot c, which is (0, 3)

For the gradient, rise over run equals fraction numerator negative 3 over denominator 5 end fraction. So plot a point 3 down and 5 right. There isn't space on the grid to repeat this so join the points with a straight line. Extend the line to the edges of the grid.

y=3x-1 and 3x+5y=15, IGCSE & GCSE Maths revision notes

Parallel Lines

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 y equals m x plus c, as they will have the same value of m (gradient)
    • y equals 3 x plus 7 and y equals 3 x minus 4 are parallel
    • y equals 2 x plus 3 and y equals 3 x plus 3 are not parallel
    • y equals 4 x plus 9 and y equals 4 x plus 9 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 y equals m x plus c will be parallel to a line in the form y equals m x plus d, where m is the same for both lines
    • If c equals d then they would be the same line and therefore not parallel
  • If you are asked to find the equation of a line parallel to y equals m x plus c, you will also be given some information about a point that the parallel line, y equals m x plus d passes through; open parentheses x subscript 1 space comma space y subscript 1 close parentheses
  • You can then substitute this point into y equals m x plus d and solve to find d

Worked example

Find the equation of the line that is parallel to y equals 3 x plus 7 and passes through (2,1)

As the gradient is the same, the line that is parallel will be in the form: 

y equals 3 x plus d

Substitute in the coordinate that the line passes through: 

1 equals 3 open parentheses 2 close parentheses plus d

Simplify: 

1 equals 6 plus d

Subtract 6 from both sides: 

negative 5 equals d

Final answer: 

bold italic y bold equals bold 3 bold italic x bold minus bold 5

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Daniel I

Author: Daniel I

Expertise: Maths

Daniel has taught maths for over 10 years in a variety of settings, covering GCSE, IGCSE, A-level and IB. The more he taught maths, the more he appreciated its beauty. He loves breaking tricky topics down into a way they can be easily understood by students, and creating resources that help to do this.