Year 9/GCSE: Simultaneous
Equations
Dr J Frost (jfrost@tiffin.kingston.sch.uk)
Last modified: 22nd September 2014
Starter
Sketch the line with 8
equation 𝑥 + 2𝑦 = 1
6
4
2
-10
-8
-6
-4
-2
2
4
6
8
-2
-4
Click to sketch
10
How many solutions for x and y?
Hint: Think about the line
representing 𝑥 = 3
For 𝒙
For 𝒚
𝑥=3
1?
∞?
𝑥2 = 4
2?
∞?
𝑥+𝑦 = 9
∞?
∞?
𝑥+𝑦 =9
𝑥−𝑦 =1
1?
1?
8
By using graphical methods,
solve the simultaneous
equations:
𝒙+𝒚= 𝟕
𝟐𝒙 − 𝒚 = −𝟏
But why does finding the
intersection of the lines give
the solution?
1. The line for each
equation represents all
the points (x,y) for which
the equation is satisfied.
-10
-8 ?at the
-6
-4
2. Therefore,
intersection(s), this gives
the points for which
both equations are
satisfied.
Bro Tip: To sketch a straight line, just
pick two values of 𝑥. If we’re
sketching 2𝑥 − 𝑦 = −1, say we pick
𝑥 = 2, then 4 − 𝑦 = −1 and thus
𝑦 = 5. Choose another value of 𝑥
and connect up.
6
Solution:
𝒙 = 𝟐, 𝒚
?= 𝟓
4
2
-2
2
4
6
8
-2
-4
Click to sketch
-6
10
Test Your Understanding
Copy the axis provided, and sketch the given lines on them*.
Hence solve the simultaneous equations.
𝑦
Q1
𝑥+𝑦 =4
𝑦 = 2𝑥 − 2
𝑦
Q2
5
5
4
4
3
3
2
2
𝑥 = 2, 𝑦 = 2
1
𝑦 = −3𝑥 + 5
𝑦−𝑥 =1
𝑥 = 1, 𝑦 = 2
1
1
2
3
4
5
𝑥
1
2
3
4
* Remember that the easiest way is to pick two points and join up, e.g. when 𝑥 = 0 and when 𝑦 = 0.
5
𝑥
www.wolframalpha.com
Thinking graphically…
For two simultaneous equations, when would we have…
0 solutions for 𝑥 and 𝑦?
Lines are parallel but not
the same.
?
Infinitely many solutions
for 𝑥 and 𝑦?
Lines are the same.
e.g.
𝑥 + 𝑦? = 2
2𝑥 + 2𝑦 = 4
Exercise 1
1
For each of the following, sketch axis for 𝑥 from
0 to 6 and 𝑦 from 0 to 6. Sketch the two lines on
your axis and use them to estimate the solution
to the simultaneous equations.
a
𝑥+𝑦 =5
𝑥 + 2𝑦 = 7
𝒙 = 𝟑, 𝒚 = 𝟐
3
a
?
b
𝑥=3
𝑥+𝑦 =5
𝒙 = 𝟑, 𝒚 = 𝟐
b
2𝑥 + 3𝑦 = 11
3𝑥 + 2𝑦 = 9
𝒙 = 𝟏, 𝒚 = 𝟑
𝑥−𝑦 =5
2𝑥 + 𝑦 = 4
𝒙 = 𝟑, 𝒚 = −𝟐
?
c
?
2
𝑥+𝑦 =2
𝑥 − 3𝑦 = −8
𝒙 = −𝟏, 𝒚 = 𝟑
?
?
c
For each of the following, sketch axis for 𝑥 from
-5 to 5 and 𝑦 from -5 to 5. Sketch the two lines
on your axis and use them to estimate the
solution to the simultaneous equations.
3𝑥 − 2𝑦 = 10
𝑥 + 4𝑦 = 1
?
𝒙 = 𝟑, 𝒚 = −
Consider the simultaneous equations:
𝑦 = 𝑎𝑥 + 3
𝑦 = 𝑏𝑥 + 𝑘
where 𝑎, 𝑏 and 𝑘 are constants. By thinking
about the lines corresponding to the equations,
under what conditions will we have:
a) Infinitely many solutions for 𝑥 and 𝑦?
𝒂 = 𝒃, 𝒌 = 𝟑
b) No solutions:
𝒂 = 𝒃, 𝒌 ≠ 𝟑
?
?
N
𝟏
𝟐
Given that:
𝑦 = 𝑥2
𝑥+𝑦 =1
Sketch suitable lines to estimate the solutions
to these simultaneous equations.
𝒙 = −𝟏. 𝟔𝟏, 𝒚 = 𝟐. 𝟔𝟐
𝒙 = 𝟎. 𝟔𝟏𝟖, 𝒚 = 𝟎. 𝟑𝟖𝟐
?
Three methods of solving simultaneous equations
by substitution
graphically
by elimination
METHOD #2: Solving by Elimination
By either adding or subtracting the equations, we can ‘eliminate’ one of the variables.
2𝑥 + 𝑦 = 6
3𝑥 − 𝑦 = 9
1
2
5𝑥
= 15 1 + 2
?Obtain by substituting
𝑥=3
your known 𝑥 into one
𝑦=0
of the two equations.
4𝑥 + 𝑦 = 6
6𝑥 + 𝑦 = 4
2𝑥
= −2
𝑥
= ?−1
𝑦
= 10
1
2
2
-
1
Bro Tip: I strongly urge you to
number your equations. This
becomes crucial when you have
three equations/three
unknowns, so that you can
indicate which equations you
are combining.
Solving by Elimination
5𝑥 + 2𝑦 = 13
2𝑥 + 2𝑦 = 4
𝑥 = 3, 𝑦? = −1
2𝑥 + 3𝑦 = 5
5𝑥 − 2𝑦 = −16
𝟒𝒙 + 𝟔𝒚 = 𝟏𝟎
𝟏𝟓𝒙 − 𝟔𝒚 = −𝟒𝟖
𝟏𝟗𝒙
=?−𝟑𝟖
𝒙
= −𝟐
𝒚
=𝟑
1
2
1 + 2
You can solve in 2
different ways:
•Eliminating 𝑥.
•Eliminating 𝑦.
Test Your Understanding
𝑥 + 3𝑦 = 10
2𝑥 − 5𝑦 = −2
𝑥 = 4,?𝑦 = 2
𝑥 − 𝑦 = −6
3𝑥 − 2𝑦 = −13
𝑥 = −1,? 𝑦 = 5
3𝑥 + 2𝑦 = 19
2𝑥 + 3𝑦 = 6
𝑥 = 9, 𝑦? = −4
If you finish quickly:
𝑥2 − 𝑧 − 𝑦2
=8
𝑥 2 + 𝑧 + 𝑦 2 = 10
−𝑥 2 + 𝑧 − 𝑦 2 = −12
𝟐𝒙𝟐 = 𝟏𝟖 → 𝒙 = ±𝟑
−𝟐𝒚𝟐 = −𝟒 → ?𝒚 = ± 𝟐
𝟐𝒛 = −𝟐 → 𝒛 = −𝟏
1 + 2
1 + 3
2 + 3
Exercise 2
1 Solve the following by
substitution.
a 𝑥 + 2𝑦 = 4
3𝑥 − 2𝑦 = 4
𝒙 = 𝟐, 𝒚 ?
=𝟏
b 2𝑥 − 𝑦 = 7
5𝑥 − 𝑦 = 16
𝒙 = 𝟑, 𝒚 = −𝟏
?
c
2 a 4𝑥 − 3𝑦 = 15
2𝑥 + 2𝑦 = −3
𝟑
𝒙 = , 𝒚 = −𝟑
𝟐
?
b 𝑥 + 6𝑦 = 3
d
?
N1
?
3
Two cats and a dog cost
£91. Three cats and two
dogs cost £159. How
much does a cat cost?
£23
?
If using textbook: GCSE Rayner (Old Edition) Page 105 - Exercise 28
𝑎2 + 𝑏 2 = 30
𝑎2 − 𝑏 2 = 20
𝒂 = ±𝟓, 𝒃 = ± 𝟓
[Cayley] Mars, his wife Venus and grandson
Pluto have a combined age of 192. The ages
of Mars and Pluto together total 30 years
more than Venus’ age. The ages of Venus and
Pluto together total 4 years more than Mars’s
age. Find their three ages.
Hint: You can form 3 equations with 3 unknowns
c 5𝑥 − 4𝑦 = 23
?
Solve:
?
3𝑥 − 3𝑦 = −5
𝟐
𝒙 = −𝟏, 𝒚 =
𝟑
4𝑥 − 3𝑦 = 18
𝒙 = 𝟑, 𝒚 = −𝟐
?
4
?
Mars = 94, Venus = 81, Pluto = 17
N2
[Maclaurin] Find all integer values that
satisfy the following equations:
𝑥 2 + 𝑦 2 = 𝑥 − 2𝑥𝑦 + 𝑦
𝑥 2 − 𝑦 2 = 𝑥 + 2𝑥𝑦 − 𝑦
Adding:
𝟐𝒙𝟐 = 𝟐𝒙
𝒙 = 𝟎 𝒐𝒓 𝟏
𝟐
When 𝒙 = 𝟎, 𝒚 = 𝒚 → 𝒚 = 𝟎 𝒐𝒓 − 𝟏
When 𝒙 = 𝟏, 𝟏 + 𝒚𝟐 = 𝟏 − 𝟐𝒚 + 𝒚
𝒚𝟐 = −𝒚
𝒚 = −𝟏 𝒐𝒓 𝟎
Thus 𝒙, 𝒚 = 𝟎, 𝟎 , 𝟎, −𝟏 , 𝟏, −𝟏 , 𝟏, 𝟎
?
Harder GCSE Exam Question
The graph shows two points
(1,7) and (3,175) on a line with
equation:
𝒚 = 𝒌𝒂𝒙
(3,175)
(1,7)
Determine 𝑘 and 𝑎 (where 𝑘
and 𝑎 are positive constants).
Answer:
a = 5, k =?1.4
Three methods of solving simultaneous equations
by substitution
graphically
by elimination
METHOD #3: Solving by Substitution
We currently have two equations both involving two variables.
𝟑𝒙 − 𝟐𝒚 = 𝟎
𝟐𝒙 + 𝒚 = 𝟕
Perhaps we could put one equation in terms of 𝒙 or 𝒚, then substitute this expression
into the other.
2x + y = 7
y = 7?– 2x
3x – 2y = 0
3x – 2(7-2x) = 0
3x – 14 + 4x = 0
7x – 14 = 0
?
7x = 14
x=2
Then y = 3
Why do you think we chose this
equation to rearrange?
Your go…
Solve for 𝑥 and 𝑦, using substitution.
2𝑥 + 𝑦 = 5
𝑥 + 3𝑦 = 5
Answer:
x = 2,?y = 1
3𝑥 − 2𝑦 = 16
𝑥+𝑦=2
Answer:
𝒙 = 𝟒, 𝒚? = −𝟐
Exercise 3
Use substitution only to solve the following simultaneous equations.
A
1 a
𝑥 + 2𝑦 = 5
2𝑥 + 3𝑦 = 8
𝒙 = 𝟏, 𝒚 = 𝟐
2
𝑥
3𝑦
?
B
3𝑥 + 2𝑦 = 3
𝒙 = −𝟏, 𝒚 = 𝟑
The angle at 𝐴 is 12° greater than the
angle at 𝐶. Find 𝑥 and 𝑦.
𝟏
𝒙 = 𝟔𝟒, 𝒚 = 𝟏𝟕
𝟑
?
c
?
?
3
d 𝑎 + 4𝑏 = 6
8𝑏 − 𝑎 = −3
𝟏
𝒂 = 𝟓, 𝒃 =
𝟒
?
e 5𝑐 − 𝑑 − 11 = 0
4𝑑 + 3𝑐 = −5
𝟑𝟗
𝟓𝟖
𝒄=
,𝒅 = −
𝟐𝟑
𝟐𝟑
?
C
𝑥
b −2𝑥 + 𝑦 = 5
2𝑥 + 𝑦 = 5
𝑥 + 3𝑦 = 5
𝒙 = 𝟐, 𝒚 = 𝟏
5
Magnus wants to buy 80 Ferraris, some
yellow and some red. He must spend
the whole of the £20m of his weekly
pocket money. He buys 𝑦 yellow
Ferraris at £40k and 𝑟 red Ferraris at
£320k. How many Ferraris of each type
did he buy?
𝒚 + 𝒓 = 𝟖𝟎
𝟒𝟎𝒚 + 𝟑𝟐𝟎𝒓 = 𝟐𝟎 𝟎𝟎𝟎
𝒓 = 𝟔𝟎, 𝒚 = 𝟐𝟎
?
£13 £19 £17
4
What is the
cost of a cat?
£1
?
[Cayley] James, Alison and Vivek go into a
shop to buy some sweets. James spends
£1 on four Fudge Bars, a Sparkle and a
Chomper. Alison spends 70p on three
Chompers, two Fudge Bars and a Sparkle.
Vivek spends 50p on two Sparkles and a
Fudge Bar. What is the cost of a Sparkle?
Sparkle = 15p
?
N1
[Maclaurin] Solve the simultaneous
equations:
𝑥+𝑦 =3
𝑥3 + 𝑦3 = 9
𝒙 = 𝟏, 𝒚 = 𝟐
𝒙 = 𝟐, 𝒚 = 𝟏
?
(You must have proved algebraically, using
substitution, that these are the only solutions)
N2
[Maclaurin] Solve:
𝑥4 − 𝑦4 = 5
𝑥+𝑦 =1
𝟑
𝟏
𝒙 = ,𝒚 = −
𝟐
𝟐
?
(Hint: If after substitution you end up with a cubic
equation, you can sometimes factorise it by
factorising the first two terms and the last two terms
first separately)
Three methods of solving simultaneous equations
SECRET LEVEL
by matrices
by substitution
graphically
by elimination