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Transcript
GCSE: Solving Linear Equations
Skipton Girls’ High School
Objectives: (a) Solve equations, including with unknowns on both sides
and with brackets.
(b) Form equations from context (with emphasis on quality of written
communication, e.g. "Let x be...").
KEY TERMS
This is an example of a:
3
2
Term
3 + 2
Expression
2
5
+1=2
A term is a product of numbers
and variables
?
(no additions/subtractions)
An expression is composed
? of one or more
terms, whether added or otherwise.
Equation
An equation says that the
? expressions on the
left and right hand side of the = have the same
value.
STARTER
The perimeter of this
work of art is 32.
By trial and error (or any
other method), find .
3 + 1
9−
=3
?
If we added all four sides of the painting to
get the perimeter, we’d have:
3 + 1 + 3 + 1 + 9 −  + 9 − 
= 4 + 20
And we’re told the perimeter is 32, so
 +  = . We’ll see today how to
‘solve’ equations like this so we can find .
Equations must always be ‘balanced’
We already know that the ‘=’ symbol means each side
of the equation must have the same value.
=4
+2
2 a
2 4
=
 + 2= 6
If we added something to one side of the equation,
what do we have to do with the other side?
+2
Equations must always be ‘balanced’
If we tripled the load on one side of the scales,
what do we have to do with the other side?
=4
×3
a
4
a a
4 4
=
×3
3 = 12
Solving
!To solve an equation means that we find the value of
the variable(s).
3 + 1
4 + 20 = 32
9−
Strategy: To get  on its own on
one side of the equation, we
gradually need to ‘claw away’
the things surrounding it.
Solving
Bro Note: You can probably see the answer to this in your
head because the equation is relatively simple, but this full
method is crucial when things become more complicated
4 + 20 = 32
?
-20
?
-20
?
4 =
12
?
4
Strategy: Do the opposite
operation to ‘get rid of’ items
surrounding our variable.
+4
4?
? 3
=
Bro Tip: Many students find writing these
operations between each equation helpful to
remind them what they’re doing to each side, but
you’ll eventually want to wean yourself off these.
3

6
-4?
?
3
?
×6



Test Your Understanding
3 − 5 = 13
?
+5
?
+5
3 = 18
?
?
3
?
3
?
=
6
When the solution is not a whole number
4 + 6 = 18
?
-4
-4?
6 = 14
?
?
6
6?
14 7
= ? =
16 3
Your Go…
3 = 20 + 4
−17 = 4
? 
=−

Bro Note: In algebra,
we tend to give our
answers as fractions
rather than decimals
(unless asked).
And NEVER EVER EVER
recurring decimals.
Dealing with Fractions

3 + = 28
5
?
-3
-3?

? = 25
5
?
×5
?
×5
 = ?125
Exercise 1
Solve the following equations,
showing full working.
1
2
3
4
5
6
 − 4 = 10
2 + 3 = 9
5 − 4 = 36
9 − 2 = 61
9 = 1 + 4
8 + 3 = 75
7
3 = 7
8
5 + 2 = 11
9
8 − 2 = 3
10
3 + 10 = 7
11
5 + 3 = 4
12
7 + 23 = 11
13
14 + 9 = 3
 =  ?
= ?
= ?
= ?
= ?
= ?

=
?


=
?


=
?


=
?


=− ?


=− ?


=− ?

14
15
16
17
18
19
20
21
22
23
24
N

=5
 =  ?
7

+3=8
 =  ?
4

−1=5
 =  ?
2

1+ =7
 =  ?
3

+3= 4
= ?
5

+8=5
 = −?
4

5= +9
 = −?
6
2
3+
=7
 =  ?
5
5

?
− 3 = 10
=
6

3

5+
=3
=− ?
4

6

11 =
+9
=− ?
7

6
+3

5
3=
+9 =− ?
8

What happens if variable appears on both sides?
5 + 3 = 2 + 9
What might our strategy be?
Collect the variable terms (i.e. The terms involving a) on one side of the
equation, and the ‘constants’ (i.e. The
? individual numbers) on the other
side.
What happens if variable appears on both sides?
Strategy? Collect the variable terms on the side of the equation where
there’s more of them (and move?constant terms to other side).
Let’s move the ‘’
terms to the left
(as 5 > 2) and the
constants to the
right.
5 + 3 = 2 + 9
This is to get rid
of the constant
term on the left.
5 = 2 + 6
-3
-3
?
?
-2a
?
-2a
3 = 6
?
?
3
?
3
 =? 2
We could
have done
these two
steps in
either order.
More Examples
11 − 4 = 2 − 13
3 + 4 = 8 − 5
9
=
5
 = −1?
Way we’d have
previously done it…
2 = −3
2?
2
=
=−
−3
3
5 = 3 − 3
Both methods are valid,
but I prefer the second
– it’s best to avoid
dividing by negative
numbers, and is less
likely to lead to error.
?
Or where we put  term on
side where it’s positive:
5 + 3 = 3
3 = −2
?2
=−
3
Test Your Understanding
3 − 3 =  + 5
 = 
?
=
3 − 5 = 5 + 2
− = 
 ?
=−

Dealing with Brackets
If there’s any brackets, simply expand them first!
2 2 + 3 = 9
 +  = 
 = 
?
=

3 − 4 2 − 3 = 7
 −  +  = 
 −  = 
?
 = 
=
Test Your Understanding
7 3 − 1 = 21 + 14
 −  =  + 
 = 
?
=
5−2 +2 =4−3 2−
 −  −  =  −  + 
 −  = − + 
 = 
?

=

Exercise 2
Solve the following.
1
2
3
4
5
6
7
8
9
10
11
12
13
3 =  + 4
6 = 4 − 4
5 + 3 = 3 + 7
8 − 3 = 6 + 7
10 + 3 = 7 − 3
 =2−
2 = 9 − 
9 = 99 − 2
 + 6 = 2 − 6
7 + 1 = 9 + 5
5  − 2 = 30
2 +1 =9+
4  − 1 = 3 − 1
14
3 −2 =3+
15
5 2 + 3 = 20
 =?
 =?−
 =?
 =?
 =? −
 =?
 =?
 =?
 =?
 =?−
 =?
 =?
 =?

 =?


 =?

16
17
18
19
20
21
22
N

10 − 5  − 2 = 3 + 4
 =?

9 − 4 = 4 − 9
 =?−

2 3 + 1 = 3 + 2 2 − 1
 = ?−


3 + 2 3 + 1 = 7 + 
 =?

2− −2 =− 2−
 =?
6+3 −2 =5 +4
 =?−
 − 2 2 −  = 3  − 2 −   =?−
1
3
1

 +  − 2 = 2 − 3
 =?
2
4
3

GCSE: Forming and Solving
Equations
Skipton Girls’ High School
RECAP :: Forming/Solving Process
Worded problem
[JMC 2008 Q18] Granny swears that she is getting younger. She has
calculated that she is four times as old as I am now, but remember that 5
years ago she was five times as old as I was at that time. What is the sum of
our ages now?
Stage 1: Represent
problem algebraically
Stage 2: ‘Solve’ equation(s)
to find value of variables.
Let  be my age and 
be Granny’s age.
 − 5 = 5 − 25
 = 5 − 20
5 − 20 = 4
 = 20
∴  = 80
 = 4
 − 5 = 5( − 5)
We previously learnt how to form expressions given a worded context.
We’ll learn how to actually solve these equations formed now! We’ll first
focus on problems where the expressions have already been specified.
Example
The angles of a triangle are as
pictured. Determine .
 + 

Step 1: Think of a sentence which would have the
word “is” in it. Write each part of sentence as an
algebraic expression, with “is” giving =.
 − 
Step 2: Solve!
Sentence with is“is” in
it?
“Sum of angles
180°”
Expr
 + 10 +
21?
+  − 50 = 180 Expr 2?
4 − 40 = 180
4 = 220 Solve!
 = 55
Another Example
+4
3

10
The rectangle and triangle have the same
area. Determine the width of the rectangle.
“Area “is”
of triangle
is area of rect”
sentence?
Expr
Expr 2?
3  1?
+ 4 = 5
3 + 12 = 5
12 = 2
 =Solve!
6
Therefore width of
rectangle is 6 + 4 = 10
Check Your Understanding
1 The following diagram shows the angles
of a quadrilateral. Determine .
3
2 + 10
2 + 20
120
“Total angle
is 360”
Expr 2?
Expr
1?
 +  +  +  +  +  = 
 +  = 
 = Solve!

 = °
2 The area of the triangle is 1 more than the
+4
5
N

12
area of the parallelogram. Determine .
Area1?
of 
Expr
is 1 more than area of par
 =  +
 +  +  = 
 = Solve!
Expr 2?
+
[JMO 1999 A9] Skimmed milk contains 0.1% fat and pasteurised whole milk contains
4% fat. When 6 litres of skimmed milk are mixed with  litres of pasteurised whole
milk, the fat content of the resulting mixture is 1.66%. What is the value of ?
Fat content of skimmed + Fat content of pasteurised = fat content of mixture
?
6 × 0.001
+ 0.04
= 0.0166  + 6
→
=4
Exercise 3
(See printed sheet)
1 The ages of three cats are ,  + 4 and 2 − 3.
7 [JMC 1998 Q18] The three angles of a
Their total age is 21. Determine .
 +  = 
=
triangle are  + 10 °, 2 − 40 °,
3 − 90 °. Which statement about the
triangles is correct? It is:
A right-angled isosceles
B right-angled, but not isosceles
C equilateral
D obtuse-angled and isosceles
E none of A-D
Solution: C
?
2 Three angles in a triangle are , 2 + 20° and
7 − 10°. What is ?
?
 = °
3 Two angles on a straight line are  + 30° and
2 − 60°. What is ?
?
 = °
?
4 The perimeter of this rectangle is 44. What is ?
Solution: 7
2 + 3
?
8
The following triangle is isosceles.
Determine its perimeter (by first
determining ).
−2
5 The area of this rectangle
is 48. Determine .
?
=
3 + 10
+5
+5
4
6 An equilateral triangle has lengths
3 + 6, 5, 5. What is ?
=
?
5 − 50
?
 +  =  −  →  = °
Perimeter =  +  +  = 
Exercise 3
9
(See printed sheet)
[JMO 2013 A7] Calculate the value of 
in the diagram shown?
Solution: 36
?
10 In the following diagram,
 = ,  = , ∠ = 7 and
∠ = 9. Determine .

7

9


 +  +  = 
?
=
11
A very large jug of  litres of orange
squash is 10% concentrate (the rest
water). When 5 litres of concentrate
is added the jug is now 12%
concentrate.
a. Form an equation in terms of
 (by considering the
amount of concentrate we
have).
.  +  = ?
. ( + )
b. Hence determine .
.  +  = .  + . 
?
.  = . 
 =  
Tougher Examples
[JMC 2013 Q7] After tennis training, Andy collects twice as
many balls as Roger and five more than Maria. They collect
35 balls in total. How many balls does Andy collect?
Let  be the number of balls Roger collects.
Then Andy collects  balls and Maria collects  − .
Total balls collected:
 +  + 
? −  = 
 −  = 
=
So Andy collected  ×  =  balls.
[TMC Regional 2014 Q9] In a list of seven consecutive numbers a
quarter of the smallest number is five less than a third of the largest
number. What is the value of the smallest number in the list?
Let smallest number be . Then numbers are ,  + ,  + ,  + ,  + ,  + ,  + 


Therefore   =   +  − 


 = ?+  − 



 =  →  = 

So smallest number is 36.
Test Your Understanding
1 The length of the rectangle is three
times the width. The total perimeter is
56m. Determine its width.
Let  be the width of the rectangle.
Then the length is .
Then the perimeter is:
 +  +? +  = 
 = 
=
∴ The width is 7m.
Bro Reminder: You
should usually start
with “Let …”
2 In 4 years time I will be 3 times as old as I was 10 years ago.
How old am I?
Let  be my age.
Then  +  =   − 
+=
?  − 
 = 
 = 
Exercise 4 (Maths Challenge Questions)
1 The sum of 5 consecutive numbers
is 200. What is the smallest
number?
Solution: 38 (a quick non-algebraic
method is to realise the middle
? of the five
number is the average
numbers, i.e. 40)
2
3
In 5 years time I will be 5 times as
old as I was 11 years ago. Form a
suitable equation, and hence
determine my age.
 +  =   − 
 =  ?
In 6 years time I will be twice as old
as I was 8 years ago. Determine my
age.
+= −
?
 = 
4
I have three times as many cats as Alice
but Bob has 7 less cats than me. In total
we have 56 cats. How many cats do I
have?
Solution: 27
?
5
Bob is twice as old as Alice at the
moment. In 4 years time their total age
will be 71. What is Alice’s age now?
4 years time: Alice  +  Bob  + 
 +  = ? →  = 
6
[TMC Final 2012 Q1] A Triple Jump
consists of a hop, step and jump. The
length of Keith’s step was three-quarters
of the length of his hop and the length of
his jump was half the length of his step.
If the total length of Keith’s triple jump
was 17m, what was the length of his
hop, in metres?
Solution: 8 metres
?
Exercise 4
7
[JMO 2003 A6] Given a “starting” number, you double it
and add 1, then divide the answer by 1 less than the
starting number to get the “final” number. If you start
with 2, your final number is 5. If you start with 4, your
final number is 3. What starting number gives the final
number 4?
 + 
=
−
 +  =   − 
?

=

Exercise 4
8
[JMO 2012 A3] In triangle , ∠ = 84°; 
is a point on  such that ∠ = 3 × ∠
and  = . What is the size of ∠?
10 [JMC 2012 Q24] After playing 500
games, my success rate in Spider
Solitaire is 49%. Assuming I win every
game from now on, how many extra
games do I need to play in order that my
success rate increases to 50%?
A 1
B 2 C 5 D 10
E 50
Let  be the number of extra games
played. Then, giving 245 games were
won before:
 +  = .   + 
 = 
Solution: Let ∠ = , then ∠ = 
and ∠ =  − . Angles in :
 +  −  +  = 
 = °
−
Then ∠ =
= °
?
?
(Note: it’s easier to just exploit the fact it’s multiple
choice and try the options!)

9
[JMO 2005 A8] A large container holds 14 litres
of a solution which is 25% antifreeze, the
remainder being water. How many litres of
antifreeze must be added to the container to
make a solution which is 30% antifreeze?
Let  be the amount of antifreeze added in
litres. 25% of 14 is 3.5. Thus:
.  +  = .   + 
=
?
N1
[JMO 2008 A9] In the diagram,  is the
bisector of angle .
Also  =  and  = . What is
the size of angle ?
Let ∠ = . Filling in the angles
using the information we find
angles in  are ,  − 
and  − . This gives  = ° .
∴ ∠ =  −  × 
= ° (Note: this is not intended to
?
be a full proof!)
Exercise 4
N2 [JMO 2010 A10] Inn the diagram, 
and  are parallel.  =  =  =
 and  =  = . Find the size
of angle .
60°

 − 60°
60°
300
− 2


 − 60°

Since  and  are parallel, ∠ and
∠ are cointerior so add to °.
If we let ∠ = , we can eventually
find the angles as pictured.
 −  +  +  −  +  = 
 = ° ∴ ∠ =  −  = °
?
N3
[JMO 2013 B2] Pippa thinks of a number. She adds
1 to it to get a second number. She then adds 2 to
the second number to get a third number, adds 3
to the third to get a fourth, and finally adds 4 to
the fourth to get a fifth number.
Pippa’s brother Ben also thinks of a number but he
subtracts 1 to get a second. He then subtracts 2
from the second to get a third, and so on until he
has five numbers.
They discover that the sum of Pippa’s five
numbers is the same as the sum of Ben’s five
numbers. What is the difference between the two
numbers of which they first thought?
Let  be Pippa’s first number. Then her numbers
are ,  + ,  + ,  + ,  + . Sum is  +
.
Let  be Ben’s first number. Then his numbers are
,  − ,  − ,  − ,  − . Sum is  − 
We’re told  +  =  −  and the
difference between their two starting numbers is
 − . Rearranging:
 +  =  − 
 −  = 
∴−=
?
Exercise 4
(See printed sheet)
N4 [JMO 2005 B4] In this figure  is a straight
line and  =  = . Also,  = .
Find the size of ∠.
(Full proof needed)
Full proof:
Let ∠ = 
Then ∠ =  (base angles of isosceles  are equal)
∠ =  (base angles of isosceles  are equal)
∠ =  (exterior angle of triangle  is sum of two interior angles)
∠ =  (base angles of isosceles  are equal)
? the angles in :
The angles in a triangle sum to °. Using
 +  +  +  = °
 = °
 = °
(We could have also used the angles in )