Academic Skills Advice
Straight Lines
A straight line graph is made up of lots of 𝑥 and 𝑦 pairs known as co-ordinates or points.
A point consists of an 𝑥 value and its corresponding 𝑦 value and is written in a bracket.
E.g. (3, 5) means that when 𝑥 = 3, 𝑦 = 5.
For example, consider the straight line 𝒚 = 𝟒𝒙 − 𝟑.
The equation (𝑦 = 4𝑥 − 3) tells us the relationship between 𝑥 and 𝑦.
You can choose any value of 𝑥 so there are an infinite number of pairs:
Let’s try some:
When 𝒙 = 𝟏, 𝑦 = 4(1) − 3, ∴ 𝒚 = 𝟏,
When 𝒙 = 𝟐, 𝑦 = 4(2) − 3, ∴ 𝒚 = 𝟓,
When 𝒙 = 𝟑, 𝑦 = 4(3) − 3, ∴ 𝒚 = 𝟗,
When 𝒙 = 𝟒, 𝑦 = 4(4) − 3, ∴ 𝒚 = 𝟏𝟑,
pair = (1, 1)
pair = (2, 5)
pair = (3, 9)
pair = (4, 13)
The above pairs (or co-ordinates) can be plotted on a graph and
joined up to make a straight line.
Often we just want an idea of what the line will look like so that we can sketch it instead of
plotting lots of pairs, and so we consider the general equation of a straight line.
The General Equation:
The general equation of a straight line is: 𝑦 = 𝑚𝑥 + 𝑐. We can tell 2 things about the
straight line from the equation:
𝑦 = 𝑚𝑥 + 𝑐
The gradient of the line
the 𝑦-intercept
(where it crosses the 𝑦-axis)
Example:
Look at the straight line equation: 𝒚 = 𝟑𝒙 + 𝟐. We know (just by looking) that this line will
have a gradient of 3 and will cross the 𝑦-axis at 2, so will look like this:
𝑦
A gradient of 3 means
that every time it goes
along 1 it goes up 3, so
it is quite steep.
Crosses the
𝑦-axis at 2
𝑥
© H Jackson 2011 / ACADEMIC SKILLS
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More about Gradients
A positive gradient is uphill:
A negative gradient is downhill:
A gradient of zero (0) is flat:
The gradient is found by dividing how much the line goes up (𝑦) by how much it goes
along (𝑥). We say “the change in 𝑦 over the change in 𝑥”.




A gradient of 1 means go along 1 and up 1.
A gradient of 2 means go along 1 and up 2 so it is steeper,
a gradient of 8 means go along 1 and up 8 so it’s getting very steep.
1
1
A gradient of 2 means go along 1 and up 2 so it is shallow…… etc
Examples:
(let the gradient = m)
m=3
m=1
1
m= 2
m = −1
m=0
1
m = −2
More about the axis-intercepts:
The 𝑦-intercept can easily be seen from the equation but you need to do a little bit of
rearranging to find the 𝑥-intercept.
This just leaves the
number on its own.
Remember:
the line crosses the 𝒚-axis when 𝒙 = 𝟎
the line crosses the 𝒙-axis when 𝒚 = 𝟎
Example:
Look at the straight line equation: 𝒚 = −𝟐𝒙 + 𝟖.
We can also get this by substituting
𝑥 = 0 into the equation.
𝒚-intercept: 𝑦 = 8 (the number on it own)
𝒙-intercept:
When 𝑦 = 0, 𝑥 =? (we need to do some rearranging here)
Replace the 𝑦 with a 0 and rearrange to find 𝑥
0 = −2𝑥 + 8
2𝑥 = 8
𝑥=4
We know that the line crosses the 𝒚-axis at 8 and the 𝒙-axis at 4 (and the gradient is −2)
© H Jackson 2011 / ACADEMIC SKILLS
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Sketching a straight line:
We can use all of the above information to sketch a straight line. When asked to sketch a
straight line, there are 3 things that it is useful to know:



The gradient
The 𝑦-intercept
The 𝑥-intercept
Example:

Sketch the line 𝒚 = 𝟐𝒙 − 𝟒
We can see that the gradient is 2 and it crosses the 𝑦-axis at −4. We have 2 out of the 3
things straight away.
It crosses the 𝒙-axis when 𝒚 = 0,
So replace 𝒚 with 0 in the equation:
0 = 2𝑥 − 4
2𝑥 = 4
𝑥=2
We now have all the information we need to sketch the line:
Gradient = 2
𝑦-intercept = −4
𝑥-intercept = 2
𝑦
𝑦 = 2𝑥 − 4
2
𝑥
−4
Plot both intercepts on
your axes then join
them with a straight line.
© H Jackson 2011 / ACADEMIC SKILLS
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Finding the Equation of a Straight Line
Sometimes you may be given 2 points on a line and asked to find the equation of the line.
Remember that the general equation of a straight line is 𝒚 = 𝒎𝒙 + 𝒄 and so the 2 pieces of
information you need to know to be able to write the equation are the gradient and the 𝒚intercept.
Gradient =
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑦
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑥
=
𝑦2 −𝑦1
𝑥2 −𝑥1
𝒚-intercept: substitute one of the points in and solve.
Examples:

Find the equation of the straight line passing through the points (2, 12) and (3, 17).
If we call the
1st
point 1 and the
2nd
(𝒙𝟏 , 𝒚𝟏 )
(𝒙𝟐 , 𝒚𝟐 )
(2, 12) and (3, 17)
point 2 then we have:
First find the gradient:
Gradient =
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑦
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑥
=
𝑦2 −𝑦1
𝑥2 −𝑥1
=
17−12
3−2
=
5
1
=5
We now know that the line has a gradient of 5 and so the equation looks like this:
𝒚 = 𝟓𝒙 + 𝑪
Next find 𝑪:
To complete the equation we need to work out 𝐶. To do this substitute one of the points in
(whichever you find easiest). In this example I’ll choose the 1st point (2, 12).
𝑦 = 5𝑥 + 𝐶
Substitute point in (replace 𝑥 with 2 and 𝑦 with 12): 12 = 5(2) + 𝐶
12 − 10 = 𝐶
𝐶=2
We now have the equation of the straight line: 𝑦 = 5𝑥 + 2
© H Jackson 2011 / ACADEMIC SKILLS
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
Find the equation of the straight line passing through the points (𝟏, −𝟒) and (𝟐, −𝟏).
Gradient =
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑦
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑥
=
𝑦2 −𝑦1
𝑥2 −𝑥1
=
−1−(−4)
2−1
=
3
1
=3
So far we know that the equation looks like this: 𝒚 = 𝟑𝒙 + 𝑪
To complete the equation we need to work out 𝐶. We need to substitute one of the points
in and I’ll choose the 1st point again (1, -4).
Substitute point in:
𝑦 = 3𝑥 + 𝐶
−4 = 3(1) + 𝐶
−4 − 3 = 𝐶
𝐶 = −7
We now have the equation of the straight line: 𝑦 = 3𝑥 − 7
N.b. if you are given the gradient and one point then the question is easier as you don’t
need to do the 1st part (finding the gradient), just start from the substitution stage.
Example:

Find the equation of the straight line with gradient −𝟐 and passing through the
point (1,1)
We are given the gradient so we know that the equation looks like this: 𝒚 = −𝟐𝒙 + 𝒄
To complete the equation we need to work out 𝐶. To do this we substitute the point in:
Substitute point in:
𝑦 = −2𝑥 + 𝑐
1 = −2(1) + 𝑐
1+2=𝑐
𝑐=3
We now have the equation of the straight line: 𝑦 = −2𝑥 + 3
© H Jackson 2011 / ACADEMIC SKILLS
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