GCSE: Straight Line Equations
Dr J Frost (jfrost@tiffin.kingston.sch.uk)
Last modified: 3rd September 2014
GCSE specification:
 Understand that an equation of the form y = mx + c corresponds to a straight line graph
 Plot straight line graphs from their equations
 Plot and draw a graph of an equation in the form y = mx + c
 Find the gradient of a straight line graph
 Find the gradient of a straight line graph from its equation
 Understand that a graph of an equation in the form y = mx + c has gradient of m and a y intercept
of c (ie. crosses the y axis at c)
 Understand how the gradient of a real life graph relates to the relationship between the two
variables
 Understand how the gradients of parallel lines are related
 Understand how the gradients of perpendicular lines are related
 Understand that if the gradient of a graph in the form y = mx + c is m, then the gradient of a line
perpendicular to it will be -1/m
 Generate equations of a line parallel or perpendicular to a straight line graph
y
What is the equation of
this line?
And more importantly,
why is it that?
4
3
2
1
x
-5
-4
-3
-2
-1
0
1
2
3
4
5
-1
-2
? 2
𝑥=
The line -3represents
all points which
satisfies -4the
equation.
□ “Understand that
an equation
corresponds to a
line graph.”
6
y
4
Starter
A
D
3
F
C
2
B
1
E
G
x
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
-1
-2
H
-3
-4
What is the equation of
each line?
Equation of a line
 Understand that an equation of the form 𝑦 = 𝑚𝑥 + 𝑐
corresponds to a straight line graph
The equation of a straight line is 𝑦 = 𝒎𝑥 + 𝒄
gradient
y-intercept
Gradient using two points

Given two points on a line, the gradient is:
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑦
𝑚=
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑥
1, 4
5, 7
2, 2
(3, 10)
𝑚 = 3?
(8, 1)
?
𝑚 = −2
(−1, 10)
8
𝑚 = −?
3
Gradient from an Equation
 Find the gradient of a straight line graph from its equation.
𝑦 = 1 − 2𝑥
Putting in form 𝒚 =
𝒎𝒙 + 𝒄:
𝒚 = −𝟐𝒙 + 𝟏
Gradient is -2
?
2𝑥 + 3𝑦 = 4
Putting in form 𝒚 =
𝒎𝒙 + 𝒄:
𝟑𝒚 = −𝟐𝒙 + 𝟒
𝟐 ? 𝟒
𝐲=− 𝒙+
𝟑
𝟑
Gradient is −
𝟐
𝟑
Test Your Understanding
Find the gradient of the line with equation 𝑥 −
2𝑦 = 1.
𝟐𝒚 = 𝒙 − 𝟏
𝟏
𝟏
𝒚= 𝒙−
𝟐 ? 𝟐
𝟏
𝒎=
𝟐
Exercise 1
1
2
Determine the gradient of the lines
which go through the following
points.
a
3,5 , 5,11
b
−1,0 , 4,3
c
2,6 , 5, −3
d
4,7 , 8,10
e
f
g
h
1,1 , −2,4
3,3 , 4,3
4, −2 , 2, −4
−3,4 , 4,3
𝒎 = 𝟑?
𝟑
𝒎= ?
𝟓
𝒎 = −𝟑
?
𝟑
𝒎= ?
𝟒
𝒎 = −𝟏
?
𝒎 = 𝟎?
𝒎 = 𝟏?
𝟏
𝒎 = −?
𝟕
Determine the gradient of the lines
with the following equations:
a 𝑦 = 5𝑥 − 1
𝒎 = 𝟓?
𝒎 = −𝟏
b 𝑥+𝑦=2
?
𝒎=𝟐
c 𝑦 − 2𝑥 = 3
?
d 𝑥 − 3𝑦 = 5
e 2𝑥 + 4𝑦 = 5
f
2𝑦 − 𝑥 = 1
g 2𝑥 = 3𝑦 − 7
3
𝟏
𝒎= ?
𝟑
𝟏
𝒎 = −?
𝟐
𝟏
𝒎= ?
𝟐
𝟐
𝒎= ?
𝟑
A line 𝑙1 goes through the points
(2,3) and 4,6 . Line 𝑙2 has the
equation 4𝑦 − 5𝑥 = 1. Which
has the greater gradient:
𝟑
𝟓
𝒎𝟏 =
𝒎𝟐 =
𝟐
? 𝟒
So 𝒍𝟏 has greater gradient.
Drawing Straight Lines
y
4
Sketch the line with equation:
𝑥 + 2𝑦 = 4
3
2
1
x
-5
-4
-3
-2
-1
0
1
2
3
4
5
-1
-2
Bro Tip: To sketch a line, just work out
any two points on the line. Then join up.
Using 𝑥 = 0 for one point and 𝑦 = 0 for
the other makes things easy.
-3
-4
□ “Plot and draw a
graph of an
equation in the
form y = mx + c”
6
y
Test Your Understanding 4
Sketch the line with equation:
𝑥 − 3𝑦 = 3
3
2
1
x
-5
-4
-3
-2
-1
0
1
2
3
4
5
-1
-2
-3
-4
□ “Plot and draw a
graph of an
equation in the
form y = mx + c”
6
Finding intersection with the axis
𝑦
𝑥
When a line crosses the 𝑦-axis:
𝒙=
?𝟎
When a line crosses the 𝑥-axis:
𝒚=
?𝟎
The point where the line crosses the:
Equation
𝒚-axis
𝒙-axis
𝑦 = 3𝑥 + 1
0,1
1
− ?, 0
3
𝑦 = 4𝑥 − 2
0, −2
?
1
?, 0
2
1
𝑦 = 𝑥−1
2
0, −1
2,0
?
?
?
Equation given a gradient and point
The gradient of a line is 3. It goes through the point (4, 10). What
is the equation of the line?
𝒚 = 𝟑𝒙 − 𝟐
? determined)
Start with 𝒚 = 𝟑𝒙 + 𝒄 (where 𝒄 is to be
Substituting: 𝟏𝟎 = 𝟑 × 𝟒 + 𝒄
Therefore 𝒄 = −𝟐
The gradient of a line is -2. It goes through the point (5, 10). What
is the equation of the line?
𝒚 = −𝟐𝒙 + 𝟐𝟎
?
Test Your Understanding
1
2
Determine the equation of the line which has gradient 5 and goes through
the point 7,10 .
𝒚 = 𝟓𝒙
? − 𝟐𝟓
Determine the equation of the line which has gradient −2 and goes through
the point 3, −2 .
𝒚 = −𝟐𝒙
? +𝟒
1
3
Find the equation of the line which is parallel to 𝑦 = − 2 𝑥 + 3 and goes
through the point 6,1
𝟏
𝒚 = −? 𝒙 + 𝟒
𝟐
Equation given two points
A straight line goes through the points (3, 6) and (5, 12). Determine
the full equation of the line.
Gradient:
3
Equation:
𝒚 = 𝟑𝒙
? −𝟑
(5,12)
?
(3,6)
A straight line goes through the points (5, -2) and (1, 0). Determine
the full equation of the line.
(5, -2)
Gradient:
-0.5
Equation:
𝒚 = − ?𝒙 +
?
(1,0)
𝟏
𝟐
𝟏
𝟐
Exercise 2
1 Determine the points where the
following lines cross the 𝑥 and 𝑦 axis.
1
𝑦 = 2𝑥 + 1
0,1 , − , 0
2
2
𝑦 = 3𝑥 − 2
0, −2 , , 0
3
5
2𝑦 + 𝑥 = 5
0, , 5,0
2
?
?
?
2
Using suitable axis, draw the line with
equation 2𝑥 + 𝑦 = 5.
𝑦
5
?
5
2
3
A line has gradient 8 and goes
through the point 2,10 . Determine
its equation.
𝒚 = 𝟖𝒙 − 𝟔
A line has gradient −3 and goes
through the point 2,10 . Determine
its equation.
𝒚 = −𝟑𝒙 + 𝟏𝟔
?
4
𝑥
?
5 Determine the equation of the line parallel
to 𝑦 = 6𝑥 − 3 and goes through the point
3,10 .
𝒚 = 𝟔𝒙 − 𝟖
?
6 Determine1 the equation of the line parallel
to 𝑦 = − 𝑥 + 1 and goes through the
3
point −9,5 .
𝟏
𝒚=− 𝒙+𝟐
𝟑
?
7 Determine the equation of the lines which
go through the following pairs of points:
3,5 , 4,7
𝒚 = 𝟐𝒙 − 𝟏
4,1 , 6,7
𝒚 = 𝟑𝒙 − 𝟏𝟏
−2,3 , 4, −3 𝒚 = −𝒙 + 𝟏
𝟐
0,3 , 3,5
𝒚= 𝒙+𝟑
𝟑
𝟓
4, −1 , 2,4 𝒚 = − 𝒙 + 𝟗
𝟐
?
?
?
?
?
y
4
m = -1/3
?
3
m = 1/2
?
2
1
m=3?
x
-5
-4
-3
-2
-1
0
1
2
3
4
5
-1
m = -2?
-2
-3
-4
Find the gradients of
each pair of
perpendicular lines.
What do you notice?
6
Perpendicular Lines

If two lines are perpendicular, then the gradient of one is the
negative reciprocal of the other.
1
𝑚1 = −
𝑚2
Or:
𝑚1 𝑚2 = −1
Gradient
Gradient of Perpendicular Line
1
−?
2
1
?3
2
−3
1
4
5
2
7
7
5
−
-4
?
1
−?
5
7
?2
5
−?
7
Example Problems
Q1
A line is goes through the point (9,10) and is perpendicular to another line with
equation 𝑦 = 3𝑥 + 2. What is the equation of the line?
𝟏
𝒚 − 𝟏𝟎 = −? 𝒙 − 𝟗
𝟑
Q2
A line 𝐿1 goes through the points 𝐴 1,3 and 𝐵 3, −1 . A second line 𝐿2 is
perpendicular to 𝐿1 and passes through point B. Where does 𝐿2 cross the x-axis?
𝟓, 𝟎
?
Q3
Are the following lines parallel, perpendicular, or neither?
1
𝑦= 𝑥
2
2𝑥 − 𝑦 + 4 = 0
𝟏
𝟏
Neither. Gradients are and 𝟐. But ×?𝟐 = 𝟏, not -1, so not perpendicular.
𝟐
𝟐
Exercise 3
1 A line 𝑙1 goes through the indicated point and
is perpendicular to another line 𝑙2 . Determine
the equation of 𝑙1 in each case.
𝟏
2,5
𝑙2 : 𝑦 = 2𝑥 + 1 𝒍𝟏 : 𝒚 = − 𝒙 + 𝟔
𝟐
𝟏
−6,3 𝑙2 : 𝑦 = 3𝑥
𝒍𝟏 : 𝒚 = − 𝒙 + 𝟏
𝟑
1
0,6
𝑙2 : 𝑦 = − 𝑥 − 1 𝒍𝟏 : 𝒚 = 𝟐𝒙 + 𝟔
2
1
−9,0 𝑙2 : 𝑦 = − 𝑥 + 1 𝒍𝟏 : 𝒚 = 𝟑𝒙 + 𝟐𝟕
3
𝟏
10,10 𝑙2 : 𝑦 = −5𝑥 + 5 𝒍𝟏 : 𝒚 = 𝒙 + 𝟖
𝟓
4
𝑙
?
?
?
?
?
2
𝐴 2,5 𝐵 4,9
Find the equation of the line which passes through B,
and is perpendicular to the line passing through both
A and B.
𝟏
𝒚 = − 𝒙 + 𝟏𝟏
𝟐
?
3
Line 𝑙1 has the equation 2𝑦 + 3𝑥 = 4. Line 𝑙2 goes
through the points (2,5) and (5,7). Are the lines
parallel, perpendicular, or neither?
𝟑
𝟐
𝒎𝟏 = −
𝒎𝟐 =
𝟐
𝟑
𝒎𝟏 𝒎𝟐 = −𝟏 so perpendicular.
?
𝑥
Determine the equation of the line 𝑙.
𝟏
𝒚=− 𝒙+𝟓
𝟑
?
5
𝑦
𝑙
𝑥
Determine the equation of the line 𝑙.
Known point on 𝒍:
𝟐, 𝟎
So equation of 𝒍:
𝟏
𝒚= 𝒙−𝟏
𝟐
?
GCSE specification:
 Understand that an equation of the form y = mx + c corresponds to a straight line graph
 Plot straight line graphs from their equations
 Plot and draw a graph of an equation in the form y = mx + c
 Find the gradient of a straight line graph
 Find the gradient of a straight line graph from its equation
 Understand that a graph of an equation in the form y = mx + c has gradient of m and a y intercept
of c (ie. crosses the y axis at c)
 Understand how the gradient of a real life graph relates to the relationship between the two
variables
 Understand how the gradients of parallel lines are related
 Understand how the gradients of perpendicular lines are related
 Understand that if the gradient of a graph in the form y = mx + c is m, then the gradient of a line
perpendicular to it will be -1/m
 Generate equations of a line parallel or perpendicular to a straight line graph
Two last things…
Distance between two points
Midpoint of two points
?5
5,9
(3,6)
3
(𝟒, 𝟕.
?𝟓)
(3,6)
Just find the average of 𝑥 and
the average of 𝑦.
(7,9)
4
Find 𝑥 change and 𝑦 change to
form right-angled triangle.
Then use Pythagoras.
Past Exam Questions
See GCSEPastPaper_Solutions.pptx
GCSERevision_StraightLineEquations.docx