Slides: GCSE Straight Line Equations
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Transcript Slides: GCSE Straight Line Equations
GCSE: Straight Line Equations
Dr J Frost ([email protected])
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