Straight Line Graphs y = mx + c | Edexcel GCSE Maths: Foundation Revision Notes 2017

Finding Equations of Straight Lines

What is the equation of a straight line?

The general equation of a straight line is y = mx + c where
- m is the gradient
  - The slope of the line
- c is the y-intercept
  - The value where it cuts the y-axis
y = 5x + 2 is a straight line with
- gradient 5
- y-intercept 2
y = 3 - 4x is a straight line with
- gradient -4
- y-intercept 3

How do I find the equation of a straight line from a graph?

Find the gradient by drawing a triangle and using
- - Positive for uphill lines, negative for downhill
Read off the y-intercept from the graph
- Where it cuts the y-axis
Substitute these values into y = mx + c

What if no y-intercept is shown?

If you can't read off the y-intercept
- Find any point on the line
- Substitute it into the equation
- Solve to find c
For example, a line with gradient 6 passes through (2, 15)
- The y-intercept is unknown
  - Write y = 6x + c
- Substitute in x = 2 and y = 15
  - 15 = 6 × 2 + c
  - 15 = 12 + c
- Solve for c
  - c = 3
- The equation is y = 6x + 3

What are the equations of horizontal and vertical lines?

A horizontal line has the equation y = c
- c is the y-intercept
A vertical line has the equation x = k
- k is the x-intercept
For example
- y = 4
- x = -2

Worked example

(a)

Find the equation of the straight line shown in the diagram below.

Graph of a straight line with negative gradient

Find m, the gradient
Identify any two points the line passes through and work out the rise and run

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

Finding the equation of a straight line from a graph

The rise is 4
The run is 8

Calculate the fraction $\frac{rise}{run}$

$\frac{rise}{run} = \frac{4}{8} = \frac{1}{2}$

The slope is downward (downhill), so it is a negative gradient

gradient, $m = - \frac{1}{2}$

Now find the y-intercept
The line cuts the y-axis at 5

y-intercept,

\begin{array}{rcl} c & = & 5 \end{array}

Substitute the gradient, m, and the y-intercept, c, into y = mx + c

$y = - \frac{1}{2} x + 5$

(b)

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

Substitute m = 3 into y = mx + c
Leave c as an unknown letter

$y = 3 x + c$

Substitute x = 5 and y = 4 into the equation
Solve the equation to find c

$\begin{array}{rcl} 4 & = & 3 \times 5 + c \\ 4 & = & 15 + c \\ - 11 & = & c \end{array}$

You now know c
Replace c with −11 to complete the equation of the line

y = 3x − 11

Drawing Linear Graphs

How do I draw a straight line from a table of values?

You may be given a table of values with no equation
Use the x and y values to form a point with coordinates (x , y )
- Then plot these points
- Use a ruler to draw a straight line through them
  - All points should lie on the line
For example
- The points below are (-3, 0), (-2, 2), ... etc

$x$	-3	-2	-1	0	1	2	3
$y$	0	2	4	6	8	10	12

How do I draw a straight line using y = mx + c?

Use the equation to create your own table of values
- Choose points that are spread out across the axes given
For example, plot y = 2x + 1 on axes from x = 0 to x = 10
- Substitute in x = 0, x = 5 and x = 10 to get y coordinates
  - Then plot those points

$x$	0	5	10
$y$	1	11	21

How do I draw a straight line without using a table of values?

Start at the y-intercept, c
Then, for every 1 unit to the right, go up m units
- where m is the gradient
- If m is negative, go down
This creates a sequence of points which you can then join up
- Be careful of counting squares if axes have different scales
  - 1 unit might not be 1 square

Examiner Tip

Always plot at least 3 points (just in case one of your end points is wrong!)

Worked example

On the same set of axes, draw the graphs of $y = 3 x - 1$ and $y = - \frac{3}{5} x + 3$ .

For $y = 3 x - 1$ , create a table of values

$x$	0	1	2
$y$	-1	2	5

Plot the points (0, -1), (1, 2) and (2, 5)
Connect with a straight line
(Alternatively, start at the y-intercept and add 3 for every 1 unit to the right)

For $y = - \frac{3}{5} x + 3$ , create a table of values
Because of the fraction, x = 5 is a good point to include

$x$	0	3	5
$y$	3	1.2	0

Plot both lines on the same set of axes

Plotting two straight lines on the same axes

Straight Line Graphs y = mx + c (Edexcel GCSE Maths: Foundation)

Revision Note

Finding Equations of Straight Lines

What is the equation of a straight line?

How do I find the equation of a straight line from a graph?

What if no y-intercept is shown?

What are the equations of horizontal and vertical lines?

Worked example

Drawing Linear Graphs

How do I draw a straight line from a table of values?

How do I draw a straight line using y = mx + c?

How do I draw a straight line without using a table of values?

Examiner Tip

Worked example

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Author: Mark