1. Year 9: Simultaneous Equations
Dr J Frost
Last modified: 13th September 2015

2. 1 ∞ 2 ∞ ∞ ∞ 1 1 𝑥=3 𝑥2=4 𝑥+𝑦 = 9 𝑥+𝑦=9 𝑥−𝑦=1
How many solutions for x and y?
Hint: Think about the line representing 𝑥=3
For 𝒙
For 𝒚 ?
𝑥=3 1 ? ∞ ? 2 ? ∞
𝑥2=4
𝑥+𝑦 = 9 ? ∞ ? ∞
𝑥+𝑦=9
𝑥−𝑦=1 ? ? 1 1

3. ? ? x + y = 7 2x – y = -1 Solution: 𝒙=𝟐, 𝒚=𝟓 8 8 6 4 2 -2 -4 -6
By using graphical methods, solve the simultaneous equations:
𝒙+𝒚=𝟕 𝟐𝒙−𝒚=−𝟏
Solution:
𝒙=𝟐, 𝒚=𝟓 ?
But why does finding the intersection of the lines give the solution?
The line for each equation represents all the points (x,y) for which the equation is satisfied.
Therefore, at the intersection(s), this gives the points for which both equations are satisfied.
x + y = 7 ?
Get students to sketch axes and tables in their books.
2x – y = -1
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.
Click to sketch

4. Test Your Understanding
Copy the axis provided, and sketch the given lines on them*.
Hence solve the simultaneous equations. 𝑦 𝑦 Q1
𝑦=2𝑥−2
𝑥+𝑦=4 Q2
𝑦−𝑥=1
𝑦=−3𝑥+5 5 4 3 2 1 5 4 3 2 1
𝑥=2, 𝑦=2
𝑥=1, 𝑦=2 𝑥 𝑥
* Remember that the easiest way is to pick two points and join up, e.g. when 𝑥=0 and when 𝑦=0.

6. Thinking graphically…
For two simultaneous equations, when would we have… ?
Lines are parallel but not the same.
0 solutions for 𝑥 and 𝑦?
Infinitely many solutions for 𝑥 and 𝑦? ?
Lines are the same.
e.g.
𝑥+𝑦=2 2𝑥+2𝑦=4

7. Exercise 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.
𝑥+𝑦=5 𝑥+2𝑦=7
𝒙=𝟑, 𝒚=𝟐
𝑥=3 𝑥+𝑦= 𝒙=𝟑, 𝒚=𝟐
2𝑥+3𝑦=11 3𝑥+2𝑦=9
𝒙=𝟏, 𝒚=𝟑
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:
Infinitely many solutions for 𝑥 and 𝑦? 𝒂=𝒃, 𝒌= 𝟑
No solutions: 𝒂=𝒃, 𝒌≠𝟑
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.
𝑥+𝑦=2 𝑥−3𝑦=−8
𝒙=−𝟏, 𝒚=𝟑
𝑥−𝑦=5 2𝑥+𝑦= 𝒙=𝟑, 𝒚=−𝟐
3𝑥−2𝑦=10 𝑥+4𝑦=1
𝒙=𝟑, 𝒚=− 𝟏 𝟐
Given that:
𝑦= 𝑥 2 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 1 3 a a ? ? b b ? ? c c ? ? 2 N ? ? ?

8. Three methods of solving simultaneous equations
by substitution
graphically
by elimination

9. 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
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. 1 2
5𝑥 =15
𝑥= 𝑦=0 ? 1 + 2
Obtain by substituting your known 𝑥 into one of the two equations.
4𝑥+𝑦=6 6𝑥+𝑦=4 1 2 ?
2𝑥 =−2 𝑥 =−1 𝑦 =10 - 2 1

10. 5𝑥+2𝑦=13 2𝑥+2𝑦=4 Solving by Elimination ? 𝑥=3, 𝑦=−1 2𝑥+3𝑦=5
5𝑥−2𝑦=−16 𝟒𝒙+𝟔𝒚=𝟏𝟎 𝟏𝟓𝒙−𝟔𝒚=−𝟒𝟖 𝟏𝟗𝒙 =−𝟑𝟖 𝒙 =−𝟐 𝒚 =𝟑
You can solve in 2 different ways:
Eliminating 𝑥.
Eliminating 𝑦. ? 1 2
1 + 2

11. Test Your Understanding
𝑥+3𝑦=10
2𝑥−5𝑦=−2
𝑥−𝑦=−6
3𝑥−2𝑦=−13
3𝑥+2𝑦=19
2𝑥+3𝑦=6 ?
𝑥=4, 𝑦=2
𝑥=−1, 𝑦=5 ?
𝑥=9, 𝑦=−4 ?
If you finish quickly:
𝑥 2 −𝑧− 𝑦 2 =8 𝑥 2 +𝑧+ 𝑦 2 =10 − 𝑥 2 +𝑧− 𝑦 2 =−12 ?
𝟐 𝒙 𝟐 =𝟏𝟖 → 𝒙=±𝟑
−𝟐 𝒚 𝟐 =−𝟒 → 𝒚=± 𝟐
𝟐𝒛=−𝟐 → 𝒛=−𝟏
1 + 2
1 + 3
2 + 3

12. Elimination by other means
[Kangaroo Grey 2013 Q6] The positive integers 𝑥,𝑦 and 𝑧 satisfy
𝑥𝑦=14 𝑦𝑧=10 𝑥𝑧=35
What is the value of 𝑥+𝑦+𝑧? ?
Multiplying the equations:
𝒙 𝟐 𝒚 𝟐 𝒛 𝟐 =𝟒𝟗𝟎𝟎 𝒙𝒚𝒛=𝟕𝟎
Thus 𝒛=𝟓, 𝒙=𝟕, 𝒚=𝟐
𝒙+𝒚+𝒛=𝟏𝟒

13. Exercise 2
[Cayley 2004 Q2] 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
Mars = 94, Venus = 81, Pluto = 17
[Cayley 2012 Q3] Three loaves of bread, five cartons of milk and four jars of jam cost £ Five loaves of bread, nine cartons of milk and seven jars of jam cost £ How much does it cost to buy one loaf of bread, one carton of milk and one jar of jam?
Solution: £2
[Maclaurin 2006 Q2] Find all integer values that satisfy the following equations:
𝑥 2 + 𝑦 2 =𝑥−2𝑥𝑦+𝑦 𝑥 2 − 𝑦 2 =𝑥+2𝑥𝑦−𝑦
Adding:
𝟐 𝒙 𝟐 =𝟐𝒙 𝒙=𝟎 𝒐𝒓 𝟏
When 𝒙=𝟎, 𝒚 𝟐 =𝒚 → 𝒚=𝟎 𝒐𝒓 −𝟏
When 𝒙=𝟏, 𝟏+ 𝒚 𝟐 =𝟏−𝟐𝒚+𝒚
𝒚 𝟐 =−𝒚 𝒚=−𝟏 𝒐𝒓 𝟎
Thus 𝒙,𝒚 = 𝟎,𝟎 , 𝟎,−𝟏 , 𝟏,−𝟏 , 𝟏,𝟎 1
Solve the following by elimination.
𝑥+2𝑦=4 3𝑥−2𝑦=4 𝒙=𝟐, 𝒚=𝟏
2𝑥−𝑦=7 5𝑥−𝑦=16 𝒙=𝟑, 𝒚=−𝟏
6𝑥−𝑦=1 2𝑥+𝑦=3 𝒙= 𝟏 𝟐 , 𝒚 =𝟐
5𝑥+𝑦=−7 3𝑥+4𝑦=6 𝒙=−𝟐, 𝒚=𝟑 3 a
4𝑥−3𝑦=15
2𝑥+2𝑦=−3
𝒙= 𝟑 𝟐 , 𝒚=−𝟑
𝑥+6𝑦=3 3𝑥−3𝑦=−5
𝒙=−𝟏, 𝒚= 𝟐 𝟑
5𝑥−4𝑦=23
4𝑥−3𝑦=18
𝒙=𝟑, 𝒚=−𝟐
Two cats and a dog cost £91. Three cats and two dogs cost £159. How much does a cat cost?
£23
Solve:
𝑎 2 + 𝑏 2 =30 𝑎 2 − 𝑏 2 =20 𝒂=±𝟓, 𝒃=± 𝟓
Solve 𝑥 2 𝑦=1, 𝑦 2 𝑧=128, 𝑧 2 𝑥=32
𝒙= 𝟏 𝟐 , 𝒚=𝟒, 𝒛=𝟖 N1 ? a ? b ? ? b N2 ? c ? c ? 4 ? N3 ? d ? 5 ? ?
[IMC 2004 Q5] The sum of two numbers is 2. The difference between them is 4. What is their product? -3 2 6 ? ?

14. Three methods of solving simultaneous equations
by substitution
graphically
by elimination

15. 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.
Why do you think we chose this equation to rearrange?
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 ?

16. 2𝑥+𝑦=5 𝑥+3𝑦=5 3𝑥−2𝑦=16 𝑥+𝑦=2 Answer: x = 2, y = 1 Answer: 𝒙=𝟒, 𝒚=−𝟐
Check Your Understanding
Solve for 𝑥 and 𝑦, using substitution.
2𝑥+𝑦=5
𝑥+3𝑦=5
3𝑥−2𝑦=16
𝑥+𝑦=2
Answer:
x = 2, y = 1
Answer:
𝒙=𝟒, 𝒚=−𝟐 ? ?

17. Exercise 3
Use substitution only to solve the following simultaneous equations. 𝑥 3𝑦 A B C
[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
[Maclaurin] Solve the simultaneous equations:
𝑥+𝑦=3 𝑥 3 + 𝑦 3 =9
𝒙=𝟏, 𝒚=𝟐 𝒙=𝟐, 𝒚=𝟏
(You must have proved algebraically, using substitution, that these are the only solutions)
[Maclaurin] Solve:
𝑥 4 − 𝑦 4 =5 𝑥+𝑦=1 𝒙= 𝟑 𝟐 , 𝒚=− 𝟏 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) 5 1 a
𝑥+2𝑦=5 2𝑥+3𝑦=8 𝒙=𝟏, 𝒚=𝟐
−2𝑥+𝑦=5 3𝑥+2𝑦=3 𝒙=−𝟏, 𝒚=𝟑
2𝑥+𝑦=5 𝑥+3𝑦=5 𝒙=𝟐, 𝒚=𝟏
𝑎+4𝑏=6 8𝑏−𝑎=−3
𝒂=𝟓, 𝒃= 𝟏 𝟒
5𝑐−𝑑−11=0 4𝑑+3𝑐=−5
𝒄= 𝟑𝟗 𝟐𝟑 , 𝒅=− 𝟓𝟖 𝟐𝟑 2 ? b
The angle at 𝐴 is 12° greater than the angle at 𝐶. Find 𝑥 and 𝑦.
𝒙=𝟔𝟒, 𝒚=𝟏𝟕 𝟏 𝟑
Gus 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 ? ? ? N1 c 3 ? ? d ? ? N2 e
£13 £19 £17 ? 4
What is the cost of a cat? £1 ? ?

18. Three methods of solving simultaneous equations
(you’ll have to wait for Further Maths A Level for this one)
SECRET LEVEL
by matrices
by substitution
graphically
by elimination